Mod-two cohomology of symmetric groups as a Hopf ring

THE MOD-TW O COHOMOLOGY RINGS OF SYMMETRIC GR OUPS CHAD GIUSTI, P AOLO SAL V A TORE, AND DEV SINHA Abstra ct. W e present a new additive basis for the mo d-tw o cohomology of symmetric groups, along with explicit rules for m ultiplication and application of Steenrod operations in that ba- sis. The key organizational tool is a Hopf ring structure introduced by Strickland and T urner. W e elucidate some of the relationships b etw ee n our approach and p revious approaches to the homology and cohomology of symmetric groups. 1. Introduction W e determine the mo d-t w o cohomology of all s y m metric groups, that is of th e disjoint union ` n B S n , as a Hopf ring. This description allo ws us to giv e the ﬁrst additiv e basis with a complete, explicit r u le for multiplicatio n in that basis. W e also giv e geometric represent ativ es f or mo d-t w o cohomology and explicitly d escrib e the action of the Steenro d algebra . Deﬁnition 1.1. A Hopf ring is a ring ob ject in the category of coalgebras. Explicitly , a Hopf ring is v ecto r space V w ith tw o multiplicat ions, one com ultiplicatio n, and an ant ip o de ( ⊙ , · , ∆ , S ) suc h that the ﬁrs t multiplica tion forms a Hopf algebra with the comultiplica tion and an tip od e, the second m ultiplication forms a bialgebra with the com ultiplica tion, and these structures satisfy the distributivit y r ela tion α · ( β ⊙ γ ) = X ∆ α = P a ′ ⊗ a ′′ ( a ′ · β ) ⊙ ( a ′′ · γ ) . W e consider only Hopf rings where all of these stru ctures are commutati v e. On th e cohomology of ` n B S n the second pro duct · is cup pr o duct, which is zero for classes supp orted on disjoint comp onen ts. The ﬁr st p rod u ct ⊙ is the transfer pro duct - see Deﬁnition 3.1 - ﬁrst stu died by Stric kland and T urner [28]. It is akin to the “ind uctio n p rod u ct” in th e r epre- sen tation theory of symmetric groups [14, 29]. The copro duct ∆ on cohomology is dual to the standard P on trjagin p r odu ct on the homology of ` n B S n . Theorem 1.2. As a Hopf ring, H ∗ ( ` n B S n ; F 2 ) is gener at e d by classes γ ℓ,n ∈ H n (2 ℓ − 1) ( B S n 2 ℓ ) , along with unit classes on e ach c omp onent. The c op r o d uct of γ ℓ,n is given by ∆ γ ℓ,n = X i + j = n γ ℓ,i ⊗ γ ℓ,j . R elations b etwe en tr ansfer pr o ducts of these gener ators ar e given by γ ℓ,n ⊙ γ ℓ,m =  n + m n  γ ℓ,n + m . The antip o de is the identity map. Cup pr o ducts of gener at ors on diﬀer ent c om p onents ar e zer o, and ther e ar e no other r elations b etwe en cup pr o ducts of g e ner ators. 1 2 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA Th us, all of the relations in the cohomolog y of symmetric groups follo w from the distrib utivit y of cup pro duct o v er transfer pro duct. Bu ilding on this presen tatio n w e giv e an additiv e basis, whic h is fairly immediate, and an explicit pr esen t ation of m ultiplicatio n rules in that basis. This additiv e b asis is represent ed graphically by “skyline diagrams” whic h are remin isce n t of Y oung diagrams. The r ule for cup pro duct is complicated but accessible, akin to r ules for m ultiplying symmetrized monomials. W e b egin the pap er by dev eloping Hopf rings which arise in algebra related to those we s tu dy , namely th at of s y m metric inv arian ts and in repr esen tations of sym m etric groups. T hough Hopf rings were in tro duced by Milgram to stud y th e homology of the sph ere sp ectrum [19] and thus the in ﬁ nite symmetric group [6], and later used to study other ring sp ectra [25 ], the Hopf rin g structure we study do es not ﬁt in to th at framework. In p artic ular it exists in cohomology rather than homology . See [28] for a lucid discussion of the relationships b et w een all of these structures. W e sh o w that these Hopf ring generators are, and th us all cohomology is, represen ted by Th om classes of linear subv arieties. W e connect with p revious w ork and identify the restriction maps in cohomology to element ary ab elian sub groups. W e use such restriction map s to study the action of the Steenro d al gebra. There is a Cartan form ula for the transf er pro duct, s o the Steenro d action on th e cohomology of symmetric groups is completely determined by that on the Hopf ring generators γ ℓ, 2 k , whic h we giv e in Theorem 8.3. W e revisit some of F esh bac h ’s calculatio ns [10] and express his cup-pr odu ct generators in terms of our Hopf ring generato rs. While the Hopf ring presentat ion of all comp onen ts is straigh tfo rwa rd, the cup ring s tr ucture for a s ingle sym metric group is s till complicat ed. W e also give our o wn in v ariant -theoretic presen tat ion. A t the end of the pap er w e sho w that Stiefel-Whitney classes for the standard representa tions can b e us ed as Hopf ring generators, forging another tie b et w een the catego ries of ﬁn ite s ets and v ector spaces. The cohomology of symmetric groups is a classical topic, d at ing b ac k to Steenro d’s [26] and Adem’s [4] stud ies of them in the con te xt of cohomolog y op erations. W e heavi ly r ely on Nak a ok a’s seminal work [21] wh ic h determined the mo d-t w o homology of s ymmetric groups. More explicit treatmen t of the cup pro duct stru cture on cohomology w as later give n at the prime tw o partially b y Hu’ng [13] and Ad em-Mc Gannis-Milgram [2] and more deﬁnitiv ely b y F esh bac h [10], usin g restriction to elemen tary ab elian subgroups and in v ariant theory . While F esh bac h’s generators for cup r ing structure are accessible, the r ela tions are giv en recursively , in in creasingl y complex form s. The Hopf ring stru cture give a compact, closed-form recursive description of all comp onents at once. It seems that it will also b e useful other primes, to other grou p s, to other conﬁgur at ion spaces, and to related spaces. Contents 1. In tro duction 1 2. Hopf r ings arising from represent ations an d inv arian ts of symmetric groups 3 3. Deﬁnition of the transf er pr odu ct 8 4. Review of homology of symmetric groups 10 5. Hopf r ing structur e thr ough generators and r ela tions 15 6. Presen tation of pro duct structures through an add itiv e basis 16 7. T opology and the in v ariant theoretic p r esen t ation 21 8. Steenro d algebra action 23 THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 3 9. Cup p rod uct generators after F eshbac h 26 10. Stiefel-Whitney generators 27 References 30 W e thank Nic k Kuhn for p oin t ing out a simpler pro of of Theorem 4.1 3 as w ell as Nic k Proudfo ot, Hal Sadofsky an d Alejandro Ad em for h elpful con v ersatio ns. The th ir d author w ould lik e to thank the Un iversities of Roma T or V ergat a, Pisa, Z u ric h, Lille, L ouv ain and Nice, and th e CIRM in F r ance and the IND AM agency in Italy for their hospitalit y . 2. Ho p f rings arising f r om repre sent a tions a nd inv ariants of symmetr ic groups In this section we iden tify some Hopf rings deﬁn ed by classical ob jects, namely rings of inv ari- an ts and represent ation rin gs of sym m etric group s , w hic h are related to th e Hopf r ing structure on the cohomology of symmetric groups. Though in v ariant theory and represen tation theory h a ve long, distinguished histories, to our kno wledge the use of Hopf r in gs to serve as a framew ork for restriction and in duction maps is n ew. Deﬁnition 2.1. Let A b e an algebra w hic h is ﬂ at o v er a ground r ing R (whic h is suppressed from notation). L et µ m,n : A ⊗ m ⊗ A ⊗ n → A ⊗ m + n denote the standard isomorphism, and let ∆ m,n denote its in v erse. Let A S = L n ( A ⊗ n ) S n , w h ic h we call the total symmetric inv arian ts of A . Deﬁne a copro duct ∆ to b e the sum of restrictions of ∆ m,n . Deﬁne a pro duct ⊙ : ( A ⊗ m ) S m ⊗ ( A ⊗ n ) S n → ( A ⊗ m + n ) S m + n as the symm et rization of µ m,n o v e r S m + n / ( S m × S n ). If A is a p olynomial algebra, deﬁ ne an an tipo de S on A ⊗ n whic h multiplies a monomial by ( − 1) k , wh ere k th e num b er of v ariables which app ear in the monomial. F or explicit cal culations with symm etric inv arian ts, whic h w e mak e th roughout th is section, w e set the follo wing notation. Deﬁnition 2.2. Let Sym( m ) denote the min imal s y m metriza tion of a monomial m , namely P [ σ ] ∈ S n /H σ · m where H is the su bgroup w hic h ﬁ x es m . Prop osition 2.3. The total symmetric i nv ariants of A , namely A S , with the pr o d uct ⊙ , its standar d pr o duct (which is zer o for elements fr om diﬀer ent su mma nds), and the c op r o duct ∆ forms a Hopf semiring. When A i s a p o lynomial algebr a, the total symmetric invariants forms a Hopf ring with the antip o de S . R emark 2.4 . This construction can b e generalized in signiﬁ ca n t w a ys. First, the r ings A ⊗ n can b e replaced by more general rings with S n action and analogues of maps µ and ∆. More generally , they could b e r eplace d by schemes, obtaining Hopf rings through regular f u nctions or p erhaps some sort of cohomology . Also, ins tead of symmetric group s other sequences of groups with inclusions G n × G m → G m + n , in particular linear groups o v e r ﬁnite ﬁ elds, can b e used. W e con ten t ours elv es her e w ith the minimum needed to tr eat cohomology of symmetric groups. Pr o o f of Pr o p osition 2.3 . The fact that the standard pro duct and ∆ form a bialgebra follo ws from the fact th at the ∆ m,n are ring h omomo rphisms. 4 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA That ⊙ and ∆ form a bialgebra is also p ossible to establish f or all of A ⊗ m ⊗ A ⊗ n , not jus t the S m × S n -in v arian ts. Firs t w e consider a 1 ⊗ · · · ⊗ a m ⊙ b 1 ⊗ · · · ⊗ b n , w hic h by deﬁn itio n the symmetrization o v e r S m + n of τ = a 1 ⊗ · · · ⊗ a n ⊗ b 1 ⊗ · · · ⊗ b m . W e c ho ose this symmetrization to b e give n b y shuﬄes. Next we app ly ∆ an d consider the ∆ m ′ ,n ′ -summand, whic h tak es the ﬁrst m ′ and last n ′ tensor factors of a give n tensor. The result of applying ∆ m ′ ,n ′ to one of the shuﬄes at hand will b e a shuﬄe of a 1 , . . . , a i and b 1 , . . . , b j tensored with a s huﬄe of a i +1 , . . . , a m with b j +1 , . . . , b n . But these pairs of smaller shuﬄes are exactly what is obtained if one ﬁr st app lies ∆ i,m − i ⊗ ∆ j,n − j to τ and then ⊙ -multiplies, establishing the result. F or d istributivit y w e start with a ∈ ( A ⊗ m + n ) S m + n and b and c in A ⊗ m and A ⊗ n resp ectiv el y . Then a · ( b ⊙ c ) is the pro duct of a w ith the symmetrization by shuﬄes of b ⊗ c . But since a is already symmetric this is equ al to the symmetrization of a · ( µ m,n b ⊗ c ) = µ m,n (∆ m,n ( a ) · b ⊗ c ). Since th ere are no other terms in the copr o duct of a wh ic h non-trivially multiply b ⊗ c , w e get that a · ( b ⊙ c ) = P ∆ a = a ′ ⊗ a ′′ a ′ · b ⊙ a ′′ ⊙ c . No w we restrict to when A is a p olynomial algebra generated by some { x i } , in whic h case the an tip ode map S multiplies a monomial by ( − 1) k , where k the n um b er of v ariables wh ich app ear in th e monomial. C onsider symmetrizations of the f orm Sym( x P ~ q ) = Sym ( x 1 q 1 x 2 q 1 · · · x p 1 q 1 x p 1 +1 q 2 · · · x p 1 + p 2 q 2 · · · x n − p k q k · · · x n q k ) , where P is the p artition P p i = n . These span the sym metric in v ariants of A so we chec k that S is an an tip od e by applyin g µ ⊙ ◦ ( S ⊗ id ) ◦ ∆ . T h e copro duct of S ym( x P ~ q ) is P P = P ′ + P ′′ Sym( x P ′ ~ q ) ⊗ Sym( x P ′′ ~ q ), wh ere P ′ and P ′′ v ary o v er partitions w ith p i = p ′ i + p ′′ i . Ap plying S × id introdu ce s a sign of ( − 1) | P ′ | to eac h term in the sum, where | P ′ | = P p ′ i . Applying ⊙ , eac h term S y m( x P ′ ~ q ) ⊙ Sym( x P ′′ ~ q ) pro duces a m ultiple of of the original s ymmetrized monomial Sym( x P ~ q ). Because Sym( x P ~ q ) had n ! P ! terms (where for a partition P ! is the p r odu ct of p i !), and Sym( x P ′ ~ q ) ⊙ Sym( x P ′′ ~ q ) has n ! P ′ ! P ′′ ! terms, this m ultiple is ( − 1) | P ′ | P ! P ′ ! P ′′ ! Sym( x P ~ q ). That S is an antip ode then follo w s from the identify P P = P ′ + P ′′ ( − 1) | P ′ | P ! P ′ ! P ′′ ! = 0 , wh ic h generalizes the familiar fact for b in omia l co eﬃci en ts when P is a singleton partition. In su mmary , what we hav e p ro v en is that b oth ( · , ∆) and ( ⊙ , ∆) d eﬁne bialgebra stru ctur es on all of ⊕ n A ⊗ n , w h ic h then restrict to in v ariants. Moreo ver, distribu tivity of · o v er ⊙ holds when multiplying something which is S n in v ariant , wh ic h means that wh en restricting to the total symmetric inv arian ts w e obtain a Hopf semiring. Finally , when A is a p olynomial algebra ( ⊙ , ∆ , S ) d eﬁne a Hopf algebra structure on the total symm etric inv arian ts, so we obtain a Hopf ring.  Deﬁnition 2.5. A Hopf r ing monomial in classes x i is one of th e form f 1 ⊙ f 2 ⊙ · · · ⊙ f k , wh ere eac h f j is a monomial und er th e · pro duct in the x i . These m onomia ls pla y a signiﬁ ca n t role in all of our examples. Example 2.6. The total symmetric inv arian ts k [ x ] S is, as a vecto r space, the direct sum of the classical rings of symm etric p olynomials ov er k . W e do not kno w wh ether this Hopf ring structur e on the d irect sum of all symm etric p olynomials h as b een considered pr evio usly . The second p rodu ct in the Hopf ring structure is the standard pro duct of s y m metric p olyno- mials, d eﬁ ned to b e zero if the num b er of v ariables d iﬀers. The copr o duct is “de-coupling” of tw o sets of v ariables follo w ed b y reindexing, so f or example ∆ 2 , 1 ( x 1 2 x 2 x 3 + x 1 x 2 2 x 3 + x 1 x 2 x 3 2 ) = ( x 1 2 x 2 + x 1 x 2 2 ) ⊗ x 1 + x 1 x 2 ⊗ x 1 2 . THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 5 The ﬁrst pro duct f ⊙ g r eind exes the v ariables of g , m ultiplies that b y f , and then symmetrizes with resp ect to S n + m / S n × S m , as can b e done with shuﬄes. A reasonable name for this pr odu ct w ould b e the shuﬄe p rodu ct. F or example ( x 1 2 x 2 + x 1 x 2 2 ) ⊙ x 1 = ( x 1 2 x 2 + x 1 x 2 2 ) x 3 + ( x 1 2 x 3 + x 1 x 3 2 ) x 2 + ( x 2 2 x 3 + x 2 x 3 2 ) x 1 = 2Sym( x 1 2 x 2 x 3 ) . Let 1 k denote the unit fu nction on k v ariables and σ n ( k ) the n th symmetric function in k v ariables. Because σ n ( k ) = σ n ( n ) ⊙ 1 k − n , as a Hopf ring symmetric fu nctions are generated by the 1 k and σ n = σ n ( n ). The σ n ⊙ -m ultiply according to the ru le σ n ⊙ σ m =  n + m n  σ n + m , a divided p o w ers algebra. Th us ov e r th e rationals only σ 1 is required to generate as a Hopf ring, w h ile o v er F p one needs all σ p i . Because of p eriod icit y of bin omia l co eﬃcien ts mo dulo p , the Hopf su b-rings generated by classes σ np i for ﬁxed i (or equiv alen tly , the quotien ts obtained by setting other symm et ric p oly- nomials to zero) are isomorphic to the full Hopf r ing of symmetric f u nctions. This isomorp hism accoun ts for some “self-similarit y” in the cohomology of symmetric groups . Irreducible Hopf ring monomials in the σ n corresp ond to s y m metrize d monomial s in the x i . That is, for p i distinct, (1) σ n 1 p 1 ⊙ σ n 2 p 2 ⊙ · · · ⊙ σ n k p k ⊙ 1 j = Sym ( x 1 p 1 · · · x n 1 p 1 x n 1 +1 p 2 · · · x n 1 + n 2 p 2 · · · x N p k ) , where N = P n i . Distributivit y in the Hopf ring structure gives rise to an indu ctive metho d to m ultiply sym metrize d monomials. This app roac h to symmetric p olynomials is f ai rly ind iﬀeren t to the classical theorem that the ring of s y m metric p olynomials in a ﬁ xed num b er of v ariables forms a p olynomial algebra. More generally , we ma y let A = k [ x 1 , . . . , x m ] in w h ic h case total symmetric in v arian ts are the direct sum of rings of symmetric polynomials in m colle ctions of v ariables. Explicitly , we tak e p olynomials in v ariables x i ,j with 1 ≤ i ≤ m and 1 ≤ j ≤ n whic h are inv arian t und er p erm utation of the second su bscripts (so that the b old s ubscripts are “ﬁxed”). Deﬁne σ i ,n to b e x i , 1 · x i , 2 · · · · · x i ,n . Example 2.7. Consider k [ x 1 , 1 , x 1 , 2 , x 2 , 1 , x 2 , 2 ] S 2 . K ey elemen ts are (1) σ 1 , 1 ⊙ 1 = x 1 , 1 + x 1 , 2 (2) σ 1 , 2 = x 1 , 1 x 1 , 2 (3) σ 2 , 1 ⊙ 1 = x 2 , 1 + x 2 , 2 (4) σ 2 , 2 = x 2 , 1 x 2 , 2 (5) σ 1 , 1 σ 2 , 1 ⊙ 1 = x 1 , 1 x 2 , 1 + x 1 , 2 x 2 , 2 . (6) σ 1 , 1 ⊙ σ 2 , 1 = x 1 , 1 x 2 , 2 + x 1 , 2 x 2 , 1 . W e then hav e the follo wing. Prop osition 2.8. The total symmetric invariants of A = k [ x i ,j ] is gener ate d as a H op f ring by unit elements and the elementary pr o ducts σ i ,n . The c opr o duct is given by ∆ σ i ,n = X j + k = n σ i ,j ⊗ σ i ,k . The ⊙ -pr o ducts ar e given by σ i ,n ⊙ σ i ,m =  n + m n  σ i ,n + m , 6 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA while ⊙ -pr o ducts b etwe en classes with diﬀer ent ℓ ar e fr e e. The c ol le ction of σ i ,n for al l i with ﬁxe d n form a p olynomial ring under the standar d pr o duct. In this presen tat ion, the standard pro duct is d ete rmined b y th e r ela tions giv en ab o v e, Hopf ring distributivity , th e fact that pro ducts of classes in diﬀeren t rings of inv arian ts are zero, and the f act that the collection of σ i ,n with ﬁxed n form a p olynomial ring. As w e discus s in Section 7, when k = F 2 these Hopf rings are isomorph ic to split quotien t-Hopf rings of the cohomolog y of symmetric groups. Pr o o f. That these symm et ric functions hav e copro ducts, ⊙ -pro ducts and ord inary pro ducts as stated is straigh tforw ard. The fact that these basic symm etric functions in eac h collecti on of v ariables are Hopf rin g generators follo ws from the fact that their asso ciate d Hopf m onomia l basis coincides with the symmetrized monomial basis for the s ymmetric p olynomials. The case of one set of v ariables is giv en in Equation 1 ab o v e . More generally we tran s la te b et w ee n these t w o b ases as follo ws. T o translate from the Hopf monomial b asis to s y m metric p olynomials p roceeds as already deﬁned. The simple monomial σ i 1 ,n p 1 · · · · · σ i k ,n p k (note that the n um b er of v ariables n for eac h symmetric function m ust b e the same to hav e the pr odu ct non-zero) is by deﬁnition ( x i 1 , 1 p 1 · · · x i 1 ,n p 1 ) · · · · · ( x i k , 1 p k · · · x i k ,n p k ) . Hopf monomials are transfer pro ducts of th ese, which by deﬁnition translate the second indices and then symmetrize. L et us denote by φ this map of sets fr om th e set of Hopf ring monomials to the set of symmetrized monomials in k [ x i ,j ]. F or example, φ ( σ 3 , 2 ⊙ σ 1 , 1 3 σ 3 , 1 3 ⊙ σ 1 , 5 7 ) is b y deﬁnition equal to Sym  ( x 3 , 1 x 3 , 2 )( x 1 , 3 3 x 3 , 3 3 )( x 1 , 4 7 · · · · · x 1 , 8 7 )  . Con v ersely , w e ma y start w ith the minimal sym metriza tion of an arbitrary monomial in the x i ,j , wh ic h if we collect terms whic h share the same second index is of th e form Sym ( x i 1 ,j 1 p 1 · x i 2 ,j 1 p 2 · · · x i m ,j k p m · · · x i ℓ ,j k p ℓ ) . Call the factor of this pro duct which shares a second ind ex j q a j -factor. W e sa y tw o j - factors are similar if th ey diﬀer on ly by relab eling of this second index. Then for eac h j - factor x i m , j p m · · · x i n , j p n and all of the j -factors which are similar to it we asso ciate the pro duct σ i m ,s p m · · · σ i n ,s p n , wh ere s is the num ber of j factors w h ic h are similar. W e then form a Hopf monomial by taking ⊙ -prod ucts of these, o v er the similarit y classes of j -factors. Let us denote b y ψ this map of sets from the set of symmetrized monomials in k [ x i ,j ] to the set of Hopf ring monomials of the total symmetric inv arian ts of k [ x i ,j ]. F or examp le, consider z = Sym  x 4 , 6 23 x 2 , 7 18 x 3 , 7 5 x 2 , 8 12 x 4 , 9 23  . The j -factors x 4 , 6 23 and x 4 , 9 23 are similar, so to them we asso ciat e σ 4 , 2 23 . T hus ψ ( z ) = σ 4 , 2 23 ⊙ σ 2 , 1 18 σ 3 , 1 5 ⊙ σ 2 , 1 12 . W e claim that φ ◦ ψ is the ident it y . I n deed, φ ( ψ ( ω )) diﬀers from ω b y the action of an element of the symmetric group wh ic h sends all of the second indices in the ﬁr st j -factor c hosen to 1 , · · · , s , all of the second indices in the second j -factor to s + 1 , · · · , and so f orth. Sin ce sym m etrize d monomials span all symm et ric p olynomials, this shows that Hopf m onomia ls in the σ i ,n span the total s y m metric inv arian ts.  THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 7 As w e will see as w ell in the case of cohomology of symmetric groups , this simple presentat ion of the Hopf ring structure b elies the fact that understand ing the standard pro duct structure alone is complicated. Example 2.9. Consider t w o sets of t w o v aria bles, namely k [ x 1 , 1 , x 1 , 2 , x 2 , 1 , x 2 , 2 ] S 2 as in Ex am- ple 2.7. T he ring of in v ariants has an add itiv e basis of σ 1 , 1 p σ 2 , 1 q ⊙ σ 1 , 1 r σ 2 , 1 s , w ith either p 6 = q or r 6 = s , along with σ 1 , 2 p σ 2 , 2 r , wh ic h we can multi ply using th e Hopf ring s tructure. Understanding this r ing in terms of generators and rela tions is already in v o lv ed. There is a fourth-degree relation, n amely 0 = ( σ 1 , 1 σ 2 , 1 ⊙ 1) 2 + ( σ 1 , 1 σ 2 , 1 ⊙ 1)( σ 1 , 1 ⊙ 1)( σ 2 , 1 ⊙ 1) + σ 1 , 2 ( σ 2 , 1 ⊙ 1) + ( σ 1 , 1 ⊙ 1) σ 2 , 2 . Indeed, the classical th eo rem that symmetric f u nctions in one set of v ariables form a p olynomial algebra is an anomaly , as ev en the simplest cases of m ultiple s et s of v ariables are quite inv olved. The structure of such rings o v er F 2 is the compu ta tional heart of Adem-McGannis-Milgram and F eshbac h’s w ork on s ymmetric group s [2, 10]. T o our kno wledge, ev en generators of such rings of in v ariant s ha v e not b een computed o v er F p with p o dd. Though we d o not apply them in this pap er, we tak e a momen t to dev elop Hopf ring s tructures on representat ion rings of symmetric group s, s ince they are direct analogues of the one we study in cohomology . In his b ook [29], Zelevinsky s ho w s that the d ir ec t sum L Rep( S n ) forms a bialgebra, und er the induction p rod uct (which he credits to Y oung) and restriction copro duct. Denote the induction pro duct, whic h take s V a rep resen ta tion of S n and W a rep resen ta tion of S m to Ind S n + m S n ×S m V ⊗ W , b y ⊙ . Denote the copro duct, which is the su m of maps whic h sends V to Res S n + m S n ×S m ( V ), by ∆ , to b e consisten t with n otation from top ology . Denote the standard pro duct in the representa tion ring, giv en by tensor pr odu ct of representa tions, by · . Let x n denote the trivial representa tion of S n , and deﬁn e an antipo de S on these b y setting ( P x i ) ⊙ ( P S ( x i )) = 0. Prop osition 2.10. O ver any gr ound ﬁeld, L n Rep( S n ) with induction pr o duct, tensor pr o duct (deﬁne d to b e zer o if r epr esentations ar e of diﬀer ent symmetric g r oups) and r estriction c opr o d- uct, forms a Hopf ring. That is, b oth induction/r estriction and S form a Hopf algebr a, ten- sor/r estriction deﬁnes a bialgebr a, and ther e is a distributivity c ondition V · ( W 1 ⊙ W 2 ) = X ∆ V = P V 1 ⊗ V 2 ( V 1 · W 1 ) ⊙ ( V 2 · W 2 ) . Sketch of pr o of. The pr oof that one obtains bialgebra structures, after un ra veling d eﬁnitions, follo ws from basic theorems on induction and restriction. Zelevinsky did not consid er th e an tip od e as part of his deﬁ nition of Hopf algebra (and w e do not kno w whether it h as b een considered b efore, or know of a more natural constru ctio n). But Zelevinsky did prov e that for complex represent ations the bialgebra deﬁ ned by in d uction pro duct and restriction copro duct alone is isomorphic to th e p olynomial algebra Z [ x 1 , x 2 , . . . ], with copro duct that ∆ x n = P x i ⊗ x j . F or a Hopf algebra whic h as an algebra is su c h a p olynomial ring, the ant ip o de S is uniqu e as given.  The c hange-of-basis b et w een the monomial basis in the x i , wh ic h corresp ond to p ermutatio n represent ations induced up f rom the trivial repr esen tation of blo c k subgroups, and the basis of irreducible represent ations is highly n on-trivia l. 8 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA Our Hopf ring structure along with Z ele vinsky’s th eo rem giv es a classically k n o w n metho d to compute tensor pro ducts of r epresen ta tions induced up from the trivia l representat ion of blo c k subgroups , that is of p erm utatio n representat ions. The Hopf monomial basis in this case coincides with the monomial basis und er ⊙ , since eac h x i 2 = x i . If P is a partition, namely p 1 + · · · + p k = n , w e let x P denote x p 1 ⊙ · · · ⊙ x p k , the p erm utation r epresen ta tion of S n induced up from the trivial represent ation of S P . F or example, for S 3 w e ha v e as an add itiv e basis: x 3 whic h is the trivial representa tion, x 1+2 whic h is the standard representa tion, and x 1+1+1 whic h is the regular representa tion. W e can compute using the Hopf ring formalism for example th at x 1+2 ⊗ x 1+2 = ( x 1 ⊙ x 2 ) · ( x 1 ⊙ x 2 ) = x 1 2 ⊙ x 2 2 + x 1 2 ⊙ (( x 1 ⊙ x 1 ) · x 2 ) = x 1 ⊙ x 2 + x 1 ⊙ x 1 ⊙ x 1 . As menti oned, th e distrib utivit y formula enco des th e ind uctio n-restriction formula, and th us leads to a classical metho d of computing tensor p rod u cts of these p ermutati on representa tions. If P and Q are partitions of n then consider an y matrix ˆ A with nonnegativ e intege r en tries su c h that the entries of i th ro w of A add u p to p i and those of the j th column of A add up to q j . Then the en tries of ˆ A form another partition of n , which we call A and sa y that A is a pro duct-reﬁnement of P and Q . F or example if P = Q = 1 + 2 then tw o p ossibilities for ˆ A are  1 0 0 1  and  0 1 1 1  . The follo wing classical theorem (see Examp le I.7.23(e) of [16 ]) is straigh tforw ard to establish using Hopf ring d istributivit y . W e state it b ecause it is a d irect analogue of our descrip tion of multiplicat ion in the cohomology of symm etric groups th rough an additiv e b asis, given in Theorem 6.8. Prop osition 2.11. If x P and x Q ar e p ermutation r epr esentations then x P ⊗ x Q ∼ = L A x A , wher e the sum is over A which ar e pr o d uct-r e ﬁnements of P and Q . More basically , we can consider the direct su m of Burnside rings of symmetric groups L A ( S n ). As u sual A ( G ) is the ring obtained by group completing the monoid of G -sets, and thus is the represent ation ring of G in the category of ﬁn ite sets. W e deﬁne m ultiplicatio n b et w een d iﬀeren t summands to b e zero. De ﬁne copro duct and induction p rod u ct analogously to ho w they we re deﬁned for repr esen tations in vecto r spaces. Once again w e obtain a Hopf ring. F or any group w e can map A ( G ) to Rep( G ) by using a G -set as a basis for a v ector s pace . Colle cting these maps give s a map L n A ( S n ) → L n Rep( S n ) which resp ects Hopf r ing structures. F or symmetric groups these maps are surj ective . 3. Def inition of t h e transfe r product The classifying sp ace for symmetric groups is often mo deled b y un ord ered conﬁguration spaces, whic h are a natural con t ext to deﬁne the second pro duct in our Hopf ring structure. Let Conf n ( X ) = { ( x 1 , . . . , x n ) ∈ X × n | x i 6 = x j if i 6 = j } . Let Conf n ( X ) = Conf n ( X ) / S n , wh ere S n acts on C onf n ( X ) by p erm uting in dices. THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 9 Deﬁnition 3.1. C onsider the follo wing maps Conf m,n ( X ) p − − − − → Conf n ( X ) × Conf m ( X ) f   y Conf m + n ( X ) . Here Conf m,n ( X ) is th e space of m + n d istinct p oints in X , m of wh ich ha v e one color and n of wh ic h hav e another. The map f f orge ts lab els, and is a cov erin g map with  n + m n  sheets. Th e map p is the pro duct of maps which pro ject on to eac h of th e t w o group s of p oin ts separately . Deﬁne the transfer pro duct ⊙ as the th e comp osite τ f ◦ p ∗ , wh ere τ f denotes the transfer map asso cia ted to f on cohomology . When X = R ∞ so that Conf n ( X ) ≃ B S n , this pro duct w as previously studied by Str ic kland and T urner [28]. In this case th e map p is a homotopy equiv ale nce, so this comp osite is essentia lly the transfer map itself. Moreo ver, in this case th e map f is homotopic to the map deﬁning the pro duct on ` n B S n . Recall that by either applying the classifying space functor to the s tand ard inclusions S n × S m ֒ → S n + m or by taking unions of un ord ered conﬁgu r ati ons in the Conf n ( R ∞ ) mo del, w e get a pr o duct on ` n B S n whic h passes to a commutati v e pro duct ∗ on its h omolo gy . Its d ual ∆ d eﬁnes a co comm utat iv e coalgebra structure on cohomology . The follo wing is immediate from Theorem 3.2 of [28]. Theorem 3.2. The tr ansfer pr o duct ⊙ along with the cup pr o duct · and the c opr o duct ∆ deﬁne a Hopf semiring structur e on H ∗ ( ` n B S n ) with c o eﬃcients in any ring. With mo d-two c o eﬃcients, the identity map gives an antip o de which deﬁnes a fu l l Hopf ring structur e. Sketch of pr o of. That cup pro duct and the coprod u ct ∆ form a bialgebra follo ws immediately from the fact that ∆ is ind uced by the co v e ring map f of Deﬁnition 3.1. The Hopf r ing d istribu- tivit y follo ws similarly from the fact that ⊙ is in d uced b y the transf er asso ciated to f . That ⊙ and ∆ f orm a bialgebra is essentia lly a double-coset formula. As Adams notes in [1] suc h form ulae usually follo w from natur al it y of transfer maps . Start with the follo wing p ull-bac k diagram of co v ering maps F Conf p,q ,r,s ( R ∞ ) e − − − − → Conf m,n ( R ∞ ) f   y g   y F Conf i,j ( R ∞ ) h − − − − → Conf m + n ( R ∞ ) , Here Con f p,q ,r,s ( R ∞ ) is deﬁned as conﬁgurations of colored p oin ts, p of w hic h hav e on e color, q of whic h hav e a second color, etc., and the ﬁrst union is o v er indices such that p + q = m , r + s = n , p + r = i and q + s = j , wh ile the second u nion is o v er those w ith i + j = m + n . All of the co v erin g maps e , f , g and h merge or forget colors. If α ∈ H ∗ ( B S n ) and β ∈ H ∗ ( B S m ) then by deﬁnition ∆( α ⊙ β ) = h ∗ ◦ g ! ( α ⊗ β ). But b ecause this is a pullback square of co v erin g spaces, this is equ al to f ! ◦ e ∗ ( α ⊗ β ), whic h is ∆( α ) ⊙ ∆( β ). Finally , for the an tip ode we need to p ass from th e space-lev e l divided p o w e rs construction W n B S n = W n E S n ⋉ S n ( S 0 ) ∧ n , which is called D S 0 , to the sp ect rum v ersion. ( See [8] for an explanation of the extension of th e functor D to sp ectra). In [28] the authors d eﬁne the an tip ode 10 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA using the additive inv erse map on the sp ectrum S 0 . But in mo d-t w o homology and cohomology , this m ap ind uces the iden tit y .  The antipo de for other co eﬃcien t sy s te ms, which we d o not consid er h ere, will b e sim ilar to that giv en for symmetric p olynomials in Prop osition 2.3, as one can see fr om the connections we dev elop in Section 7. R emark 3.3 . The transfer pro duct is ind u ced by a stable map, namely the transfer τ f : Σ ∞ Conf m + n ( X ) − → Σ ∞ Conf m,n ( X ) . The comp osite Ψ = Σ ∞ p ◦ τ f : Σ ∞ ( Conf n ( X )) → Σ ∞ (  Conf k ( X ) × Conf n − k ( X )  is a stable map inducing on homology the copro duct ∆ ⊙ dual to the transfer pro duct. Because these structures are all deﬁned b y applying cohomolog y to maps and stable maps, the generalized cohomology of s y m metric groups for an y r ing theory forms a Hopf rin g, as established in Theorem 3.2 of [28]. Str ickland in [27] u ses this to study the Mora v a E -theory of sym metric groups. The K -theory of symmetric groups is a co mpletion of the representati on ring, by the A tiy a h-Segal th eo rem [5], and the Hopf ring s tructure deﬁ n ed by Str ickland-T ur ner agrees with that of Prop osition 2.10. The cohomology and r ep resen ta tion theory of v arious t yp es of lin ea r groups ov er ﬁnite ﬁelds also form Hopf rings, as do some r ings of inv arian ts und er these groups suc h as Dic kson algebras, as do the cohomology of some s y m metric pro ducts. Using these Hopf ring structur es for further study is likely to b e fruitful. Because the fundamen tal geometry underlyin g th e tran s fer pro duct is that of taking a con- ﬁguration and partitio ning it in to t w o conﬁgurations in all p ossible wa ys, we sometimes call it the p artiti on p rod uct. Indeed, partitioning is part of th e geometry as seen through Poincar ´ e dualit y . T he usu al cup p r odu ct corresp ond s to in tersection of P oincar ´ e duals (that is, su p p orts of represen ting Thom classes), whic h means ta king the the lo cus of conﬁgurations of n p oin ts whic h satisfy the conditions deﬁn in g b oth of the t w o co cycles in question. The lo cus d eﬁning the transfer p rod uct is similar, bu t we instead r equire that some k p oin ts satisfy th e ﬁrst condition and then th e complemen tary n − k p oints satisfy the latter condition. 4. Re view o f homology of symme t ric groups W e n ow fo cus on calculations with F 2 co eﬃci en ts, recollecting standard facts to set notation. Deﬁnition 4.1. Let K b e the asso cia tiv e algebra ov er F 2 , with p r odu ct ◦ , generated by q 0 , q 1 , . . . with th e follo wing r ela tions, (Adem) F or m > n, q m ◦ q n = X i  i − n − 1 2 i − m − n  q m +2 n − 2 i ◦ q i . Giv en a sequence I = i 1 , · · · , i k of non-negativ e integ ers, let q I = q i 1 ◦ · · · ◦ q i k . Using the Adem relations, K is s panned by q I whose en tries are non-decreasing. W e call such an I admissible . If suc h an I has n o zeros we call it str o ngly admissible . THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 11 F ollo wing [7], we call K the Kudo-Araki algebra, to distinguish it from a closely related pre- sen tation usu ally called the Dy er-Lashof algebra. The algebra K is one of the main charac ters in algebraic top ology b ecause it acts on th e homology of an y inﬁnite lo op space, or more generally an y E ∞ -space (see I.1 of [9]). Deﬁnition 4.2. An action of K on a graded algebra A with pro duct denoted ∗ and grading denoted d eg is a map from K ⊗ A → A , t ypica lly written using op erational n ota tion, w ith the follo wing prop erties: • (Action) ( q i ◦ q j )( a ) = q i ( q j ( a )). • (Grading) deg q i ( a ) = 2 deg a + i . • (Squaring) q 0 ( a ) = a ∗ 2 . • (V anish ) q i (1) = 0 for i > 0. • (Cartan) F or any a, b , w e h a ve q n ( a ∗ b ) = P i + j = n q i ( a ) ∗ q j ( b ). If K acts on A we call A a K -algebra. W e d en ot e p o w e rs in su c h an algebra A by a ∗ · · · ∗ a = a ∗ n . As is standard, there is a free K -algebra functor, left adjoin t to th e forgetful functor from K - algebras to vect or spaces, with whic h we can giv e the simplest reformulatio n of Nak a ok a’s semin al result. Theorem 4.3 ([21]) . H ∗ ( ` n B S n ) , with i ts standar d pr o duct ∗ , is i somorphic to the fr e e K - algebr a gener ate d by H 0 ( B S 1 ) . Thus, as a ring under ∗ i t is isomorph ic to the p olynomial algebr a gener ate d by the nonzer o class ι ∈ H 0 ( B S 1 ) and q I ( ι ) ∈ H ∗ ( B S 2 k ) for I str ongly admissible. The second statemen t, whic h is essentiall y Nak aok a ’s f ormulation, follo ws s tr ai gh tforw ardly from the ﬁrst statemen t. W e will often abuse notation and refer to q I ( ι ) as sim p ly q I ∈ H | I | B S 2 k with | I | = i 1 + 2 i 2 + · · · + 2 k − 1 i k . W e describ e geometric representat iv es of the classes q I , and then some of their du al cohomology classes. Deﬁnition 4.4. Give n I = i 1 , i 2 , · · · , i k inductiv ely d eﬁne manifolds Orb I and map s Q I : Orb I → Conf 2 k ( R d ) where d > i ℓ for all ℓ as follo ws. • If I is empt y Orb I is a p oin t. Oth er w ise, O r b I = S i 1 × Z / 2 (Orb I ′ × Orb I ′ ), the qu otient of Z / 2 acting an tip odally on S i 1 and by p erm uting the t w o factors of Or b I ′ , w here I ′ = i 2 , · · · , i k . • Let ε = 1 4 . If I is emp t y , Q I sends O r b I to 0 = R 0 . O th erwise, Q I  v × Z / 2 ( o 1 , o 2 )  is giv en by ( v + εQ I ′ ( o 1 )) S ( − v + εQ I ′ ( o 2 )). Here w e consider v ∈ S i 1 to b e a un it v ector in R i 1 +1 . The conﬁ gu r atio n v + εQ I ′ ( o 1 ) is the conﬁguration obtained by scaling eac h p oin t in Q I ′ ( o 1 ) b y ε and then add ing v , p erh aps after either the conﬁgur ati on or v is includ ed (canonically) in to the larger of the t w o Euclidean sp aces in which they are deﬁ ned. Prop osition 4.5. The class q I in The o r em 4.3 is e qual to ( Q I ) ∗ [Orb I ] ∈ H ∗ ( Conf 2 k ( R ∞ )) , wher e [Orb I ] i s the fundamental class of Or b I . W e n ow p r esen t “linear” geometric repr esen t ativ es f or cohomology classes. 12 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA Figure 1. An illustration of q 1 , 2 ∈ H 5 ( B S 4 ). Deﬁnition 4.6. Let ˆ X b e a manifold without b oundary of dimens ion nd − m wh ich maps prop erly to Conf n ( R d ). W e deﬁn e its Thom class as follo ws. T ak e the fu ndamen tal class in mo d-t w o lo cally- ﬁnite homology of ˆ X in dimens ion n d − m , and map it to lo ca lly-ﬁnite h omolo gy of Conf n ( R d ). Apply the Poi ncar ´ e d ualit y isomorphism to obtain a class in in H m ( Conf n ( R d ); F 2 ), whic h w e call the Th om class of ˆ X . By abu se w e m ay sometimes r efer only to the image of ˆ X . If we c ho ose d large enough (greater than m ), then the r estricti on map from the cohomology of Conf n ( R ∞ ) = B S n to th at of C onf n ( R d ) is an isomorp hism in degree m . (One can dedu ce this isomorphism from calculations in homology , which f or Conf n ( R d ) is constructed from q I with i k < d .) S o th is Thom class lifts uniquely to deﬁne a class in the cohomolog y of B S n whic h we also call the Th om class. F or example, if we refer to p oints in Conf n ( R d ) as x = ( x 1 , . . . , x n ) / ∼ , then the non-zero class in H 1 ( Conf n ( R d )) for d ≥ 2 is represente d by the v ariety X of p oints su c h that some x i and x j share their ﬁ rst co ordinate. Sp eciﬁcally , let ˆ X b e the space of conﬁgurations of n p oints, tw o of whic h ha v e one color - say blac k - and the rest of whic h share another color, s uc h that th e t w o blac k p oints m ust share their ﬁrst co ordinate. The Thom class of ˆ X mappin g to th e conﬁguration space by forgetting colors is the non-trivial class in degree one, as we can see b y ev a luating it on the cycle q 1 b y intersectio n (exactly once do t w o unlab eled p oints whic h are ant ip o dal on some generic S 1 share th eir ﬁr s t co ordinate). Suc h Th om classes repr esen t Hopf ring generators of the mo d-t w o cohomology of symm etric groups. Deﬁnition 4.7. Let q ℓ · 1 denote q 1 ,..., 1 and similarly let q k · 0 ,ℓ · 1 denote q 0 ,..., 0 , 1 ,..., 1 = q 1 ,..., 1 ∗ 2 k in H 2 k (2 ℓ − 1) ( B S 2 k + ℓ ), where th ere are k zeros and ℓ ones. Let γ ℓ,n denote the linear dual to ( q ℓ · 1 ) ∗ n in th e Nak aok a monomial basis. If α ∈ H ∗ ( ` n B S n ) is a monomial in the q I w e let α ∨ ∈ H ∗ ( ` n B S n ) denote the cohomology class which ev aluates to one on α and is zero on all other mon omials. In particular γ ℓ, 2 k is q k · 0 ,ℓ · 1 ∨ . W e will use these γ ℓ,n as Hopf ring generators for th e cohomolog y of sym metric groups, th us m aking the follo wing the ﬁrst s tep in geometrically repr esen tin g this cohomology . Deﬁnition 4.8. Let Γ ℓ,n b e deﬁned as the collection of x = ( x 1 , · · · , x n 2 ℓ ) / ∼ whic h can b e partitioned in to n sets of 2 ℓ p oin ts such that all p oin ts in eac h set share their ﬁrst co ordinate. THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 13 Figure 2. An illustration th at Γ 2 , 1 in tersects Orb 1 , 1 exactly once. Γ ℓ,n is the image in Conf n · 2 ℓ ( R d ) of d Γ ℓ,n , in whic h along with the p oints there is a choic e of partition. Theorem 4.9. The c oho molo gy class γ ℓ,n is the Thom class of the variety Γ ℓ,n . W e ﬁ r st record the follo wing, wh ic h is immediate algebraically from the deﬁnition. Lemma 4.10. The c opr o duct of γ ℓ,n is given by ∆ γ ℓ,n = X i + j = n γ ℓ,i ⊗ γ ℓ,j . Pr o o f of The or em 4.9. W e start by showing that the copro duct formula of Lemma 4.10 holds for th e th e Thom class of Γ ℓ,n . W e mo del the pro d uct map b y the emb edding Conf i 2 ℓ ( R ∞ ) × Conf ( n − i )2 ℓ ( R ∞ ) → Conf n 2 ℓ ( R ∞ ) by using homeomorphism of R w ith the negativ e (resp ectiv ely p ositiv e) real num b ers to c hange th e ﬁrst co ordinates of the ﬁrst (resp ectiv ely second) given conﬁgurations and taking their un ion. T h is mod el of th e pro duct map is transv ersal to Γ ℓ,n , whose preimage in Conf i 2 ℓ ( R ∞ ) × Conf ( n − i )2 ℓ ( R ∞ ) is exa ctly Γ ℓ,i × Γ ℓ,n − i . Bec ause the Thom class of a preimage of a su bv ariet y under a transversal map is the pull-bac k of its Thom class, the desired copro duct form ula follo ws. By Nak aok a ’s calculation as stated in Theorem 4.3, the indecomp osables in homology of F n B S n under the pro duct lie in dimensions greater than n − 1 on comp onen ts in d exed by n which are p o w ers of t w o. T he pro duct map is th us surjectiv e in low er degrees, or du al ly the copro duct map is injectiv e, whic h imp lies th at we may use these copro duct formulae indu ctivel y to redu ce to showing that the Thom class of Γ ℓ, 1 is γ ℓ, 1 . Still using the copro duct form ula, along with compatibilit y of ev aluation of cohomology on h omo logy w ith the pr odu ct and copro duct, th e v alue of Γ ℓ, 1 on an y pro duct is zero. In degree (2 ℓ − 1) the only indecomp osible in h omol ogy is q ℓ · 1 . W e can c hec k immediately that Orb ℓ · 1 whic h repr esents q ℓ · 1 in tersects with Γ ℓ, 1 in exactly one p oint, and the tangen t ve ctors span the full tangen t sp ac e of the conﬁguration space as needed for tran s v e rsalit y , as w e illustrate in Figure 2.  These v ariet ies Γ ℓ,n are analogues of S c hub ert v arieties, as w e will see more precisely in Sec- tion 10. W e can u se other co ordinates, or co dimension one subs pace s, to deﬁn e Γ ℓ,n and then use 14 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA the geo metry of cup and transfer p rod ucts to understand rep resen ti ng v arieties. F or example, γ 1 , 2 γ 2 , 1 is Thom class of the su bv ariet y deﬁned by “four p oin ts whic h share their ﬁrst co ordinate and break up in to t w o groups of t w o p oints which sh are their second co ordinate,” wh ile γ 3 1 , 1 ⊙ γ 1 , 1 is the Thom class of the subv ariet y deﬁ n ed b y conﬁgurations with “t w o p oint s w hic h share th eir ﬁrst three co ordinates and another t w o which share their fourth co ordinate.” Getting b ac k to h omolo gy , on the q I the copr odu ct du al to the cup pr odu ct is classically kno wn, and th us it is determined on the en tire homology of symmetric group s b ecause of the bialgebra structure. Deﬁnition 4.11. Deﬁne a copr odu ct ∆ · on H ∗ ( ` n B S n ) by extending the form ula for I admis- sible ∆ · ( q I ) = P J + K = I q J ⊗ q K , where w hen I = i 1 , · · · , i n w e hav e that J and K range ov er partitions of the same length such that for eac h ℓ , j ℓ + k ℓ = i ℓ . This copr odu ct is more complicated than it seems at ﬁrst. Even when starting w ith an admis- sible I , the sum ab o v e is ov er all p ossible J and K . T h us to get an exp ression in the s tand ard basis, as needed for example to apply the copro duct again, one must app ly Adem and q 0 relations. The ones wh ich get used most often are the relations q 2 n +1 q 0 = 0 and q 2 n q 0 = q 0 q n . Theorem 4.12 (See for example I.2 of [9]) . Under the isomorph ism of The or em 4.3, the diagonal map on ` n B S n induc es the map ∆ · on homolo gy. On the other h and, one of our main r esults is th at the copr o duct dual to the transfer pro duct is primitive. W e originally pro v ed this geometrically , but no w b y Kuh n’s suggestion we use th e mac hinery of [8]. Theorem 4.13. The tr ansfer pr o duct is line arly dual to the primitive c opr o duct on the Kudo- Ar aki-Dyer-L a shof algebr a. Tha t is, ∆ ⊙ ( q I ) = q I ⊗ 1 + 1 ⊗ q I , wher e 1 is the non-zer o class in H 0 ( B S 0 ) . Pr o o f. Let D k ( Y ) = E S k ⋉ S k Y k , for a based sp ace Y . W e r eca ll that D ( Y ) = W k ≥ 0 D k ( Y ) is the fr ee E ∞ -space Y , so in particular D ( S 0 ) = W n ≥ 0 B S n . Similarly for a sp ectrum S we denote b y D ( S ) the free E ∞ -sp ectrum it generates. I n p artic ular D (Σ ∞ Y ) ≃ Σ ∞ D ( Y ). As mentioned in Remark 3.3, the transfer pro duct is induced by a stable map Ψ : Σ ∞ D ( S 0 ) → Σ ∞  D ( S 0 ) ∧ D ( S 0 )  . Observe th at there is an equiv alence β : D ( S 0 ∨ S 0 ) ≃ D ( S 0 ) ∧ D ( S 0 ). Recall Theorem 4.3, wh ic h s a ys that H ∗ ( D ( S 0 )) is the free K -algebra on the generator ι ∈ ˜ H 0 ( S 0 ). Sim ilarly , H ∗ ( D ( S 0 ∨ S 0 )) is the fr ee K -algebra on the t w o generators ( ι, 0) , (0 , ι ) ∈ ˜ H 0 ( S 0 ∨ S 0 ) . Clearly β ∗ ( ι, 0) = ι ⊗ 1 and β ∗ (0 , ι ) = 1 ⊗ ι . Theorem 1.5 in [8] states that Ψ can b e iden tiﬁed to D ( v ) : D ( S 0 ) → D ( S 0 ∨ S 0 ), where v : S 0 → S 0 ∨ S 0 is the p inc h map of the sphere sp ectrum. This imp lies that ∆ ⊙ is a map of K -algebras. Explicitly , b y the external Cartan formula, ∆ ⊙ q n ( a ) = X i + j = n ( q i ⊗ q j )∆ ⊙ ( a ) , THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 15 for a ∈ H ∗ ( ` n B S n ). The class ι is clearly primitive since v ∗ ( ι ) = ( ι, 0) + (0 , ι ) implies ∆ ⊙ ( ι ) = ι ⊗ 1 + 1 ⊗ ι . The external Cartan formula and the v a nishing prop erty of Deﬁnition 4.2 imply that q I is p r imitiv e for eac h I b y ind u ctio n on the length of I , completing the pro of.  W e use this theorem to quickly determine th e cohomolo gy of symm et ric groups as a Hopf ring. 5. Ho p f ring st r ucture through gener a tors an d rela tions The pr imitivi t y of the transfer copro duct coupled with some classical theo rems immediately leads to algebraic presen tations of H ∗ ( ` n B S n ). Recall from Th eorem 7.15 of Milnor and Mo ore’s standard reference [20 ] that a Hopf algebra which is p olynomial and primitive ly generated has a linear dual that is exterior, generated b y linear duals (in the monomial basis) to generators raised to p ow e rs of tw o. Theorem 4.13 implies the follo wing. Corollary 5.1. Under the tr ansfer pr o duct ⊙ alone, the c ohomolo gy of ` n B S n is an exterior algebr a, gene r ate d by ( q I ∗ 2 k ) ∨ for I str ongly admissible, or e quivalently by q I ∨ for I admissible. T o incorp orate the cup pro duct structure, we app eal to another classical theorem. As in I.3 of [9], let R [ n ] b e the span of th e q I of length n , a su b mod ule of H ∗ ( B S 2 n ). Recall that γ ℓ,m is the linear dual to ( q ℓ · 1 ) ∗ m in th e Nak aok a basis. Theorem 5.2 (Theorem I.3.7 of [9]) . The line ar dual of R [ n ] , which is an algebr a under cup pr o duct, is isomorph ic to the p olynomial algebr a gener ate d b y th e classes γ ℓ, 2 k with k + ℓ = n , which we denote D n . In Section 7 we review the classical fact that D n is canonically isomorph ic to the n th Dic kso n algebra. F or a sketc h of pro of of Theorem 5.2, it is simple to see that q 0 ,..., 0 , 1 ,..., 1 are primitive under ∆ · b ecause q 1 ◦ q 0 = 0. It is a straigh tforw ard ind uctio n to show ther e are no other primitiv es, and then a coun ting argument to show that there are no relations among the q k · 0 ,ℓ · 1 ∨ . Note ho w ev e r that b ecause of the Adem relations in the Kud o-Araki-Dy er-Lashof algebra, the pairing b et w een q I and p olynomials in γ ℓ, 2 k = ( q ℓ · 0 ,k · 1 ) ∨ is complicated. F or example, γ 1 , 2 3 = ( q 0 , 1 ∨ ) 3 = q 0 , 3 ∨ + q 2 , 2 ∨ , in p art since ∆ · q 2 , 2 includes as a term q 2 , 0 ⊗ q 0 , 2 , whic h is equ al to q 0 , 1 ⊗ q 0 , 2 . Theorem 5.2 giv es u s the last inpu t we n ee d to und erstand the cohomology of sym m etric groups as a Hopf rin g, since w e see th at und er ⊙ alone the generators are p olynomials in the γ ℓ, 2 k . Theorem 5.3. A s a Hopf ring, H ∗ ( ` n B S n ) is gener a te d by the classes γ ℓ, 2 k . The tr a nsfer pr o duct is exterior, and the antip o de map is the identity. The γ ℓ, 2 k with ℓ + k = n form a p ol ynomial ring. The stated facts along with Hopf ring d istributivit y and the f ac t that the pro ducts of classes on diﬀerent comp onents are zero determine the cup pr odu ct structur e, in particular of individu al comp onen ts. Corollary 5.4. Any c ol le ction of classes { α ℓ, 2 k } such that α ℓ, 2 k ∈ H 2 k (2 ℓ − 1) ( B S 2 k + ℓ ) p airs non- trivial ly with q k · 0 ,ℓ · 1 c onstitutes a gener ating set for H ∗ ( ` n B S n ) as a Hopf ring. In ord er to apply th e distr ib utivit y r ela tion, we need the copro duct of γ ℓ, 2 k as given by Lemma 4.10. Finally , w e compute transfer p rod ucts by taking the b inary expansion j = P 2 k i 16 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA with k i distinct, so that q ℓ · 1 ∗ j = Q q ℓ · 1 ∗ 2 k i . By Theorem 4.13 and lin ear dualit y it follo ws that γ ℓ,j = K j = P 2 k i γ ℓ, 2 k i . Because the transf er pro duct is exterior, w e obtain the follo wing. Prop osition 5.5. The tr ansfer pr o ducts of classes γ ℓ,n ar e given by γ ℓ,n ⊙ γ ℓ,m =  n + m n  γ ℓ,n + m , while tr a nsfer pr o ducts b etwe en other classes have no r elation s. Collecting T heorem 5.3, Lemma 4.10 , Theorem 5.2 and Pr oposition 5.5 yields a p r oof of Th e- orem 1.2, our ﬁrs t presen tation of the cohomology of symmetric groups as a Hopf ring. Theorem 5.3 and the fact that transfer p rodu ct is exterior leads to an additiv e basis for th e cohomology of symmetric group s. Since the γ ℓ, 2 k are Hopf ring generators, their Hopf ring mono- mials span, and an induction by num ber of transfer pr odu cts using Theorem 4.13 sho ws th ey are indep endent. Let P ( m ) denote the set of p artitio ns of m into nonnegativ e p ow ers of tw o. If P is su c h a partition, let P n denote the num b er of times 2 n o ccurs, and let D P denote N n V P n D n where D n is the p olynomial algebra on γ ℓ, 2 k with ℓ + k = n as s ta ted in Theorem 5.2. By con v en tion D 0 is the groun d ﬁeld. W e reco v er an additiv e isomorphism well-kno w n to exp erts. Prop osition 5.6. As a gr a de d ve ctor sp a c e, H ∗ ( B S n ) is isomorphic to L P ∈P ( n ) D P . It is straigh tforw ard to calculate the Poinca r ´ e p olynomial for this cohomology . W e h a ve found it unenlight ening since we could only describ e Po incar ´ e p olynomials of exterior p o w ers of p olynomial algebras u sing inclusion-exclusion metho ds. In the next section we dev elop a slightly improv ed basis, whic h requires th e follo wing. Prop osition 5.7. The classes { γ ℓ,n } such that n 2 ℓ = m gene r ate a p o lynomial su b algebr a of H ∗ ( B S m ) . Sketch of pr o of. W e start with m = 2 p whic h is co vered b y Theorem 5.2, which says th at the classes γ ℓ, 2 k with k + ℓ = p f orm a p olynomial s ubring of H ∗ ( B S 2 p ). W e then us e in induction on the n um b er of ones in th e binomial expan s io n of m and Hopf ring d istr ibutivit y to establish the general case.  By Theorem 5.3 that γ ℓ,n are Hopf r ing generators, Th eo rem 4.9 that these classes are Thom classes of linear subv arieties, and the geometric interpretatio ns of cup and transfer p rodu cts, all of the cohomology of symmetric groups is r epresen ted by suc h subv arieties. These are d eﬁned by groupings and subgroup ings of p oin ts into sets with cardinalities wh ic h are p o w e rs of tw o whic h share coord inate s. The third author b egan this inv estigation b y conjecturing that repr esen tations b y linear subv arieties w ould b e p ossible, since linear su b manifolds represent classes in ord ered conﬁguration spaces. 6. Pres ent a tion of product st r uctures through an additive bas is W e can us e our kno wledge of the Hopf ring structure on H ∗ ( ` n B S n ) to explicitly understand the cu p and tran s fer pro duct structures through additiv e bases. W e set the notational con v en tio ns THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 17 15 γ 4 , 1 14 γ 3 , 2 13 12 γ 2 , 4 11 10 9 γ 2 , 3 8 γ 1 , 8 7 γ 3 , 1 γ 1 , 7 6 γ 2 , 2 γ 1 , 6 5 γ 1 , 5 4 γ 1 , 4 3 γ 2 , 1 γ 1 , 3 2 γ 1 , 2 1 γ 1 , 1 B S 2 B S 4 B S 6 B S 8 B S 10 B S 12 B S 14 B S 16 Figure 3. Hopf ring generators of H ∗ ( ` n B S n ) throu gh B S 16 . that cup pro duct has p riorit y o v er transf er pr odu ct, so that a · b ⊙ c means ( a · b ) ⊙ c , and that exp onen ts alw a ys refer to rep eated application of cup pro duct (an easy c hoice, since tr an s fer pro duct is exterior). Let 1 m denote the un it for cup p r odu ct on comp onent m . W e pr oceed with some calculations on the ﬁr st ev en comp onent s (since the F 2 -cohomolog y of B S 2 k + 1 is isomorphic to that of B S 2 k .) As B S 2 ≃ R P ∞ its co homology is a p olynomial ring generated by γ 1 , 1 . F or H ∗ ( B S 4 ) the Hopf rin g mon omial basis consists of classes γ 1 , 1 i ⊙ γ 1 , 1 j ∈ H i + j ( B S 4 ) (which are zero if i = j ) along with p olynomials in γ 1 , 2 ∈ H 2 ( B S 4 ) and γ 2 , 1 ∈ H 3 ( B S 4 ). Using Hopf ring d istributivit y , we ha v e that ( γ 1 , 1 i ⊙ γ 1 , 1 j ) · ( γ 1 , 1 k ⊙ γ 1 , 1 ℓ ) = γ 1 , 1 i + k ⊙ γ 1 , 1 j + ℓ + γ 1 , 1 i + ℓ ⊙ γ 1 , 1 j + k , where w e n ot e one of these term s could b e zero, if either i + k = j + ℓ or if i + ℓ = j + k . In order to compute some pro ducts with γ 1 , 2 w e ha v e to use its copro duct, which by L emm a 4.10 is equal to γ 1 , 2 ⊗ 1 0 + γ 1 , 1 ⊗ γ 1 , 1 + 1 0 ⊗ γ 1 , 2 . Usin g distribu tivit y , γ 1 , 2 · ( γ 1 , 1 n ⊙ γ 1 , 1 m ) = ( γ 1 , 2 · γ 1 , 1 n ) ⊙ (1 0 · γ 1 , 1 m )+ ( γ 1 , 1 · γ 1 , 1 n ) ⊙ ( γ 1 , 1 · γ 1 , 1 m ) + (1 0 · γ 1 , 1 n ) ⊙ ( γ 1 , 2 · γ 1 , 1 m ) = γ 1 , 1 n +1 ⊙ γ 1 , 1 m +1 . In general, most terms arising from the Hopf r ing distr ibutivit y relation are zero b eca use they in v olv e m ulti plication of classes supp orted on diﬀeren t comp onen ts. The last basic p rodu cts to compute for B S 4 are γ 2 , 1 · ( γ 1 , 1 n ⊙ γ 1 , 1 m ), whic h are zero b ecause the copro duct of γ 2 , 1 is j u st γ 2 , 1 ⊗ 1 0 + 1 0 ⊗ γ 2 , 1 . App lying distributivit y rep eatedly we get that if k 6 = 0 γ 1 , 2 p γ 2 , 1 q · ( γ 1 , 1 k ⊙ γ 1 , 1 ℓ ) = ( γ 1 , 1 k + p ⊙ γ 1 , 1 ℓ + p if q = 0 . 0 if q 6 = 0 , 18 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA whic h completes an und erstanding of how to m ultiply elemen ts of our additiv e basis. The case of B S 4 is one of th e ve ry f ew in whic h it is simp ler to u nderstand the cup m ultiplicativ e structur e in terms of ring generators and relations. F r om th e m ultiplicativ e rules ju st giv en, it is a straigh tfor- w ard exercise to d educe that γ 1 , 1 ⊙ 1 2 , γ 1 , 2 and γ 2 , 1 generate the cohomology on this comp onen t, with the lone r ela tion b eing ( γ 1 , 1 ⊙ 1 2 ) · γ 2 , 1 = 0. Similarly , one can wr ite down an add itiv e basis for H ∗ ( B S 6 ), determine its m ultiplicatio n rules, and then sh o w that it is generated by γ 1 , 1 ⊙ 1 4 , γ 1 , 2 ⊙ 1 2 , γ 2 , 1 ⊙ 1 2 and γ 2 1 , 1 ⊙ γ 1 , 1 ⊙ 1 2 , with th e relation that γ 2 , 1 · ( γ 1 , 1 2 ⊙ γ 1 , 1 ⊙ 1 2 ) = 0, in agreemen t w ith the results of Ch apter VI.5 of [3]. R emark 6.1 . W e can also see relations throu gh ou r geometric repr esen t ativ es for cohomology . F or example, γ 2 , 1 · ( γ 1 , 1 ⊙ 1 2 ) is rep r esen t ed b y the subv ariet y of “four p oin ts w hic h share their ﬁrst co ordinate, tw o of wh ich share their s ec ond co ordinate.” This subv ariet y is cob ounded by “four p oints, tw o of whic h sh are their ﬁrs t co ordinate, tw o of w hic h share their ﬁ rst and second co ordinate, with th e ﬁrs t t w o having a ﬁrst co ordinate which is less than that of the s econd t w o.” In general, presentati ons in terms of generators and relations are qu ite complicated. W e in stea d understand cup and transfer p r odu cts explicitly in terms of a canonical additiv e basis. Recall the notion of Hopf rin g monomial f r om Deﬁnition 2.5. Deﬁnition 6.2. A gathered monomial in the cohomology of symmetric groups is a Hopf ring monomial in th e generators γ ℓ,n where such n are maximal or equiv alently the num b er of tr an s fer pro ducts which app ear is minimal. F or examp le, γ 1 , 4 γ 2 , 2 3 ⊙ γ 1 , 2 γ 2 , 1 3 = γ 1 , 6 γ 2 , 3 3 . Gathered monomials s u c h as the latter in w hic h no trans f er pro ducts app ear are building b loc ks for general gathered monomials. Deﬁnition 6.3. A gathered blo c k is a monomial of the form Q i γ ℓ i ,n i d i , where the pro duct is the cup pro duct. Its proﬁle is deﬁn ed to b e the collecti on of p airs ( ℓ i , d i ). Non-trivial gathered blo c ks m ust hav e all of the n um b ers 2 ℓ i n i equal, an d we call this num b er divided b y tw o the width . W e assume that the factors are ord ered from smallest to largest n i (or largest to s m all est ℓ i ), and th en note that n i = 2 ℓ 1 − ℓ i n 1 . Prop osition 6.4. A gather e d monomial c an b e written u niquely as the tr ansfer pr o duct of gath- er e d blo cks with distinct pr oﬁles. Gather e d monomials form a c anonic al additive b asis for the c ohomolo gy of ` n B S n . Represen ting gathered monomials graph ica lly is helpfu l. W e represen t γ ℓ,n b y a rectangle of width n · 2 ℓ and heigh t 1 − 1 2 ℓ , so that its area corresp onds to its degree. W e represen t 1 n b y an edge of wid th n (a heigh t-zero rectangle). A gathered b lock, whic h is a p rod uct of γ ℓ,n for ﬁxed n · 2 ℓ , is represente d by a single column of suc h rectangles, stac k e d on top of eac h other, with order wh ich do es not m at ter. A gathered monomial is represente d b y placing suc h columns next to eac h other, wh ic h we call the skyline diagram of the monomial. W e also refer to the gathered monomial basis as the skyline basis to emphasize this presen tatio n. See Figure 4 b elo w for an illustration. Deﬁnition 6.5. Let Q i γ ℓ i ,n i d i b e a gat hered b loc k, and let n 1 = P k j =1 m j b e a partition of n 1 . A partition of th is gathered blo c k in to k is d eﬁned b y the set consisting of the k blo c ks Q i γ ℓ i , 2 ℓ 1 − ℓ i m j d i . W e allo w for some m j to b e zero, in wh ic h case the corresp ondin g elemen ts of the partition will b e 1 0 . THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 19 A splitting of a gathered mon omial f 1 ⊙ · · · ⊙ f k in to t w o is a pair of gathered monomials f ′ 1 ⊙ · · · ⊙ f ′ k and f ′′ 1 ⊙ · · · ⊙ f ′′ k where eac h { f ′ i , f ′′ i } is a partition of f i in to t w o (which could b e trivial - that is, of the form { 1 0 , f i } ). Prop osition 6.6. The c o pr o duct of a gather e d monomial is giv e n b y ∆ f 1 ⊙ · · · ⊙ f k =  X f ′ 1 ⊙ · · · ⊙ f ′ k  ⊗  f ′′ 1 ⊙ · · · ⊙ f ′′ k  , wher e the sum is over al l splittings of the monomial into two. Pr o o f. T o establish the sp ecial case of gathered blo c ks - th at is, h a vin g only one f - w e u se the Hopf algebra compatibilit y of cup pro duct and P on try a gin copro duct. The copro duct ∆ of any γ ℓ i ,n i d i will corresp ond to partitions of n i in to tw o . But only for the partitions of n 1 will there b e corresp onding partitions for all n i whic h yield non-trivial classes wh en cu p p ed together. The resulting p rod ucts corresp ond to the partitions of f in to t w o. The general case follo ws from th e Hopf algebra compatibilit y of partition pr o duct and P on- try agin copro duct. Because the monomial is gathered, n o terms in the copro ducts of f i can b e equal, s o we obtain n o trivial transfer pro ducts when suc h terms are collec ted.  In terms of skylin e diagrams, the copro duct can b e un derstoo d b y in tro ducing vertica l dashed lines in the rectangles repr esen t ing γ ℓ,n , dividin g the rectangle in to n equ al p iece s. The copro duct is then giv en by divid ing along all existing columns and ve rtical d ashed lines of fu ll h eight and then p artitio ning them int o t w o to mak e t w o n ew skyline d iag rams. Deﬁnition 6.7. A partition of a gathered monomial in H ∗ ( B S m ) is a partition of eac h of its gath- ered blo c ks. T he asso ciated comp onen t partition is the partition of m giv en by the comp onents of the classes in the partition. W e deﬁne the r eﬁ nemen t of a partition of a gathered monomial in the ob vious wa y , reﬂected faithfully by the reﬁn emen t structure of th e asso cia ted comp onen t p artitio ns. A matc hing µ b et w een partitions of tw o gathered monomials is an isomorphism of their re- sp ectiv e comp onen t p artit ions. W e sa y that one matc hing r eﬁnes another if that isomorphism comm utes with inclusions of comp onen ts under reﬁn emen t . F or any gathered monomial in H ∗ ( B S m ) there is a canonical partition of m deﬁned by the comp onen ts of its constituen t gathered b loc k monomials. The asso ciated comp onen t p artiti on of a monomial p artit ion is a reﬁn emen t of th is canonical partition. W e are no w ready to describ e pro duct structur es in term s of our additive basis of gathered monomials. Theorem 6.8. The tr ansfer pr o duct of two gather e d monomials m and n is a multiple of the gather e d monomia l whose gather e d blo ck of a given pr oﬁle has width which is the sum of the widths of the blo c ks of that pr oﬁle in m and n . The multiple is zer o if and only if any of those two widths shar e some non-zer o digi t of their binary exp ansion. L et M m , n denote the set of matchings b etwe en any of the p artitions of these g ather e d monomials which ar e not a r eﬁnement of some other matching. The cup pr o duct of x and y is the sum X µ ∈ M m , n   K b,b ′ matche d by µ β µ b · b ′   , 20 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA · = + (a) ( γ i 1 , 1 ⊙ γ j 1 , 1 ) · ( γ 1 , 1 ⊙ 1 2 ) = γ i +1 1 , 1 ⊙ γ j 1 , 1 + γ i 1 , 1 ⊙ γ j +1 1 , 1 · = (b) ( γ i 1 , 1 ⊙ γ j 1 , 1 ) · γ 1 , 2 = γ i +1 1 , 1 ⊙ γ j +1 1 , 1 · = 0 (c) ( γ i 1 , 1 ⊙ γ j 1 , 1 ) · γ 2 , 1 = 0 · = + (d) ( γ 3 1 , 1 ⊙ γ 2 , 1 ⊙ 1 2 ) · ( γ 1 , 2 ⊙ 1 4 ) = γ 4 1 , 1 ⊙ γ 2 , 1 ⊙ 1 2 + γ 3 1 , 1 ⊙ γ 1 , 2 γ 2 , 1 ⊙ 1 2 Figure 4. Some compu tati ons in H ∗ ( B S 4 ) and H ∗ ( B S 8 ), expressed by b oth gathered monomials and skylin e diagrams. wher e β µ is zer o if and only i f ther e ar e two pr o ducts b · b ′ which r e su lt i n blo cks with the same pr oﬁle and whose widths have binary exp a nsion which shar e a non-zer o digit. Graphically , trans fer p rod u ct corresp onds to placing t w o column S kyline diagrams next to eac h other and merging columns with th e same constituent blo c ks, with a coeﬃcien t of zero if an y o f those column widths share a one in their dy adic expansion. F or cup pro duct, w e start with t w o column diagrams and consider all p ossible w a ys to split eac h into columns, along either original b oundaries of columns or along the vertica l lin es of full height inte rnal to the rectangles represent ing γ ℓ,n . W e th en m at c h columns of eac h in all p ossible wa ys up to automorphism, and stac k th e resulting matc hed columns to get a new s et of columns – see the Figure 4. Pr o o f of The or em 6.8. W e u se gathered blo c ks, wh ose multiplica tion is p olynomial by Pr oposi- tion 5.7 as a b ase case for an induction on the total num b er of blo c ks in m and n . View sa y m as a non -trivial transfer p rod uct of m ′ and m ′′ whic h preserves b loc ks, s o that m ′ and m ′′ eac h has f ew er blo c ks than m . The k ey is to see that eac h matc hing in M m , n coincides with some (arbitrary) partition of y into { n ′ , n ′′ } al ong with matchings of partitions of those pieces w ith partitions of m ′ and m ′′ . F rom this observ a tion, the indu ctio n f oll o ws, with the co eﬃcien t β µ accoun ting for when suc h a pro cess yields a p artit ion pro duct of some monomial in th e γ ℓ, 2 k with itself.  THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 21 Giv en that the basis of skyline diagrams is a f undamen tal cohomology basis, it wo uld b e helpfu l to understand the pairing of gathered monomials with Nak a ok a’s monomial b asis for homology . P olynomials in γ ℓ, 2 k are the fu ndamen tal case, whic h as m entioned after Theorem 5.2 pair non- trivially with the basis of q I with I of length k + ℓ . 7. Topo logy and the inv ariant th eoretic present a tion Compare the pr esen t ation f or the cohomology of symmetric grou p s as a Hopf r ing, as give n in Theorem 1.2, with the Hopf ring p resen ta tions of rings of symmetric functions, as giv en in Ex- ample 2.6 and Prop osition 2.8. Seeing classes whic h b eha v e similarly , w e obtain some immediate iden tiﬁcations of sp lit quotien t r ings of the cohomology of s ymmetric groups. Deﬁnition 7.1. Deﬁne the lev el- ℓ quotien t Hopf ring of the cohomology of symmetric groups, denoted P ℓ , to b e th e quotien t Hopf ring obtained b y setting all γ ℓ ′ ,n for ℓ ′ 6 = ℓ equal to zero. This q u otie n t map is sp lit by the sub -Ho pf ring generated by the classes γ ℓ,n . Let P ℓ [ m ] b e the sub-mo dule of P ℓ supp orted on B S m , wh ic h is an algebra un d er cup pro duct. Graphically , these su b-rings eac h consist of all skyline diagrams made from the blo c ks of one ﬁxed size. Prop osition 7.2. The level- ℓ Hopf ring P ℓ is isomo rphic as Hopf ring to th e total symmetric invariants of a k [ x ] . Thus P ℓ [ m ] i s a p o lynomial ring for any m . The pr oof is an immediate comparison of their tw o present ations. W e originally pro v ed the second p art directly from the Hopf ring pr esentati on of P ℓ [ m ], b efore realizing that we were mimic king the p roof that symmetric f u nctions form a p olynomial algebra. This identiﬁcati on has the follo wing signiﬁcan t generaliza tion. Deﬁnition 7.3. Deﬁne the scale of γ ℓ,n to b e the p rod u ct ℓ · | n | 2 , w here | n | 2 is the 2-adic v aluation of n (t hat is, the largest p ow er of tw o whic h divides n ). Deﬁne the scale - k quotien t of the cohomology of symmetric groups, d enote d Q k , to b e the qu oti en t Hopf ring obtained by setting all γ ℓ,n with either scale less than k or with ℓ > k to zero. It is isomorphic to the sub-Hopf ring generated b y γ ℓ,n with s ca le greater than or equal to k and ℓ ≤ k . Graphically , the skyline diagrams whic h are non-zero in this quotien t are those made up of blo c ks of width exactly 2 k − 1 . In the lev el- ℓ case, the sp lit su b-ring w as compatible with our additiv e basis, in th at if a sum of gathered monomials w as in th e sub-rin g then eac h m on omial w as in the su b-ring itself. That is n ot tru e in the scale- ℓ setting, where for example γ 1 , 2 3 includes a term of γ 1 , 1 3 ⊙ γ 1 , 1 2 ⊙ γ 1 , 1 . Prop osition 7.4. The sc ale- k Hopf ring Q k is isomorphic as a H opf ring to the total symmetric invariants of a p olynomial ring in k variables. Once again, the pro of is b y a direct comparison, made p ossible b y the Hopf ring approac h . The canonical isomorphism b et w een them s en ds γ ℓ,n with scale greater than or equal to k and ℓ ≤ k to th e symmetric p olynomial σ ( ℓ ) m with m = n 2 k − ℓ . Our goals in the rest of this section are t w ofold. First w e devel op the standard top olog y whic h und erlies these isomorp h isms. Then, we mo v e fr om these id en t iﬁcations of lo ca l inv arian t- theo ertic su b/quotien t r ings to the global in v arian t-theoretic description of Theorem 7.10 . 22 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA The predominant approac h to the cohomology of symm etric group s has b een through restrict- ing cohomology to that of elementa ry ab elia n sub groups. F or the follo wing, we refer to Chap- ters 3 and 4 in [3]. Let V n denote the su bgroup of ( Z / 2) n ⊂ S 2 n deﬁned by h a vin g ( Z / 2) n act on itself. If w e view this actio n as give n by linear translations on the F 2 -v ecto r s p ace ⊕ n F 2 , then we can see that the normalizer of this subgroup is isomorph ic to all aﬃne transformations of ( F 2 ) n . The W eyl group is thus GL n ( F 2 ), wh ic h acts as exp ected on the cohomolog y of V n . The in v ariants F 2 [ x 1 , . . . , x n ] GL n ( F 2 ) are known as Dic kson algebras, whic h are p olynomial on generators d k ,ℓ in dimensions 2 k (2 ℓ − 1) wh er e k + ℓ = n . As m en t ioned earlier, these Dic kson algebras together form a Hopf ring, whic h w e are curr en tl y inv esti gating. Since we base our work on Nak aok a’s h omol ogy calculatio n, our analysis of elemen ta ry ab elian subgroups in v olv e s homology as w ell as cohomology . Lemma 7.5. The image of the homolo gy of B V n in that of B S 2 n is exactly the sp a n of the q I for I admissible of length n . Pr o o f. The inclusion of V n in to S 2 n factors through the n -fold iterated wreath pro duct of Z / 2 with itself, that is Z / 2 R  Z / 2 R  · · ·  Z / 2 R Z / 2  · · ·  . A well-kno wn alternate deﬁn itio n of th e Dy er- Lashof op erations q i is th rough the homology of th e inclusion of wreath pro ducts Z / 2 R S n → S 2 n . Inductive ly , th e image of this iterated w reat h p r odu ct is give n b y length- n Kud o- Araki-Dy er- Lashof classes, s o the image of V n is con tained in th e span of such op erations. T o see that th e image of V n yields all su c h classes we compare ranks usin g the du al map in cohomology . W e cl aim that th e imag e in cohomology of the inclusion of V n in S 2 n is all of the Dic kson inv arian ts F 2 [ x 1 , . . . , x n ] GL n ( F 2 ) ∼ = F 2 [ d k ,ℓ ] with k + ℓ = n and ℓ > 0 , a fact known by Milgram [18] w hic h we share now. The standard r ep resen ta tion of S 2 n through p ermutatio n matrices giv es rise to a vecto r bundle. Because V n em b eds in S 2 n through the linea r a ction of ( F 2 ) n on itself, on passin g to a p ermutatio n r ep resen ta tion the standard repr esen t ation yields the sum of all one-dimensional real representa tions of ( Z / 2) n . Thus wh en the corresp onding bundle is pulled b ac k to B V n it splits as the su m of all p ossible line b undles. So the total Stiefel-Whitney class of this stand ard bund le in the cohomology of B S 2 n maps to Q y ∈ H 1 ( B V n ) (1 + y ), where y ranges o v er linear combinatio ns of the x i . But classical inv arian t theory identiﬁes P d k ,ℓ with the pro duct of all 1 + λ where λ v a ries o v er all linear functions in the x i . S o these Stiefel-Whitney classes map exactly to the Dic kson generators (or to zero). By Ma dsen’s calculation, T heorem I.3.7 of [9] as reco unte d in Theorem 5.2 , the linea r dual to the sp an of the length- n q I is a polynomial algebra in classes of dimension 2 k (2 ℓ − 1) with k + ℓ = n and ℓ > 0. Since the image of the cohomology of B S 2 n in that of B V n has the same rank as this p olynomial algebra, and thus as the span of q I of length n , the image in homology m ust b e all of th is span.  Because the only classes in the W eyl-in v arian t cohomology of B V n in degrees 2 k (2 ℓ − 1) are the Dic kson classes, an d the map in homolo gy send s a generator in that degree to q k · 0 ,ℓ · 1 , we hav e the follo wing. Corollary 7.6. The r estriction of γ ℓ, 2 k with k + ℓ = n to the e lementa ry ab elian sub gr oup V n is the Di ckson class d k ,ℓ . F ollo wing Quillen [23, 24], Guna w ardena-Lannes-Zarati [11] sho w ed that H ∗ ( B S n ) injects in the direct sum of the cohomology of element ary ab elian subgroup s . Lemma 7.5 giv es an alternate pro of for that theorem through the f ollo wing reﬁn emen t. THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 23 Corollary 7.7. The image of the elementary ab elian sub gr oup Q i V k j in the homolo gy of any symmetric gr oup which c ontains it (that is, of or der P 2 k j or gr e ater) i s the sp an of pr o ducts Q q I j wher e I j is of length k j . Thus, the map fr o m the homolo gy of al l elementary ab elian sub gr oups to the homolo gy of symmetric gr o ups is surje ctive. W e n ow give a top olog ical int erpretation of Prop osition 7.4. Theorem 7.8. The map fr om H ∗ ( ` n B S n ) to its image in the c oh omolo gy of ` m B V k m 2 k c oin- cides with the quotient map deﬁning the sc ale- k quotient ring Q k . Pr o o f. By C orollary 7.7, the image in h omol ogy of ` m B V k m 2 k is the su bmo d ule of pr odu cts of q I of length k . By T heorem 4.12 and Theorem 4.13, it is closed u nder the copro ducts dual to cup and transfer p rod uct. Thus the image of this map of classifying spaces in cohomology , linear du al to th is image of homology , is a quotien t of th e cohomology of symmetric group s as a Hopf r ing. Recall that γ ℓ,n = ( q ℓ · 1 ∗ n ) ∨ , so that all γ ℓ,n with either scale less than k or ℓ > k will ev a luate to zero on the image of homology . An elementa ry counting argumen t shows that this ideal, the quotien t b y wh ic h deﬁn es Q k , is as large as p ossible so that the restriction of the cohomology of symmetric groups to these elemen tary ab elian s ubgroups is exactly Q k .  W e n ow give a global in v ariant theoretic description of the cohomolog y of symmetric groups. Deﬁnition 7.9. C onsider the ring of p olynomials F 2 [ x A ], wh ere A ⊆ m = { 1 , . . . , m } . W e call A ′ a trans late of A if they are disjoint and of the same cardinalit y . A collec tion of translates is to b e m utually disjoin t. C al l a monomial Q x A i prop er if whenever some A i and A j in tersect, one is conta ined in the other, sa y A i ⊂ A j , and A j is the u nion of tr anslat es of A i Theorem 7.10. The c oh omolo gy of symmetric gr oups H ∗ ( ` n B S n ; F 2 ) is isomorphic to to the quotient of L m F 2 [ x A | A ⊂ m ] S m , with x A in de gr e e 2 # A − 1 , b y the additive submo dule c onsisting of symmetrizations of monomials which ar e not pr op er. Sketch of pr o of. W e b egin with the abstract Ho pf ring description of H ∗ ( ` n B S n ; F 2 ) giv en in Theorem 1.2, and sh o w that it is isomorphic to th e quotien t stated. Giv en s ome A = { 1 , · · · , 2 k } ⊂ m deﬁne its i th translate τ i A to b e { 1 + i 2 k , · · · , ( i + 1)2 k } . W e start to deﬁne a map betw e en H ∗ ( ` n B S n ; F 2 ) and this ring of inv arian ts b y sending γ ℓ,n to Q n i =1 x τ i A where A = { 1 , · · · , 2 ℓ } . More generally , the gathered blo c k Q γ ℓ i ,n i d i maps to the symmetrized pr odu ct where the ﬁr s t n i translates of x 1 , ··· , 2 ℓ i are r ai sed to th e d i th p o w er. Such pro ducts are prop er monomials. The transfer pro ducts of gathered blo c ks go to symm etriz ed pro ducts of suc h monomials, after reindexing so that the su bscripts corresp onding to diﬀerent gathered blo c ks are distinct. J u st as w e show ed for s y m metric functions in Prop osition 2.8, with patience we can see that all prop er symmetric monomials can, afte r reindexing, b e p ut in this form.  8. S teenrod algebra action W e no w giv e a presenta tion of the cohomology of symmetric groups as algebras o v er the mo d- t w o S tee nro d algebra A . Prop osition 8.1. The Ste enr o d squar es S q i satisfy a Cartan f ormula with r esp e ct to tr ansfer pr o duct. That is S q i ( α ⊙ β ) = P j + k = i S q j α ⊙ S q k β . 24 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA Because transfers are stable maps, they preserve Steenro d squares. So this prop osition is im- mediate from Deﬁn itio n 3.1 and the extern al Cartan form ula. Beca use there are C arta n f ormulae with r esp ect to b oth pro ducts, th e cohomology of ` n B S n is a Hopf ring ov er the Steenro d algebra. The S tee nro d algebra structure on all of the cohomology of ` n B S n is th us d ete rmined by the action on Hopf rin g generators, and w e consid er the minimal set of γ ℓ, 2 k . W e in trod u ce some notation to d escrib e the action of Steenro d squ ares on these classes, u sing the add iti v e basis of gathered monomials f or skyline diagrams f rom Section 6. Deﬁnition 8.2. • The h eight of a gathered monomial is the largest of the algebraic degrees of its gathered blo c ks. (The algebraic degree is the total num ber of Hopf r in g generators cup-m ultiplied to giv e the gathered blo c k.) • The eﬀectiv e scale of a gathered blo c k, whic h is a pro duct of γ ℓ,n ’s, is th e largest ℓ wh ich o ccurs. The eﬀectiv e scale of a gathered m onomia l is the minim um of the eﬀectiv e scales of its constituent blo c ks. • W e say a m onomia l is not f u ll w id th if it is a non-trivial tr ansfer pro duct of some monomial with s ome 1 k . Theorem 8.3. S q i γ ℓ, 2 k is the sum of al l ful l-width monomials of total de gr e e 2 k (2 ℓ − 1) + i , height one or two, and e ﬀe ctive sc ale at le a st ℓ . W e call su c h monomials the outgro wth monomials of γ ℓ, 2 k . F or examp le, S q 3 γ 2 , 4 = γ 4 , 1 + γ 3 , 1 ⊙ γ 2 , 1 γ 1 , 2 ⊙ γ 2 , 1 + γ 2 , 1 2 ⊙ γ 2 , 1 ⊙ γ 2 , 2 . W e can translate the conditions of Theorem 8.3 to our skyline diagrams, seeing that a Steenro d square on γ ℓ, 2 k is r epresen te d by the s u m of all diagrams whic h are of full width , with at most t w o b oxes stac k ed on top of eac h other, and with the width of columns delineated by any of th e v ertical lines (of fu ll heigh t) at least ℓ . The example ab ov e translates to S q 3 ( ) = + + . W e establish this theorem th r ough r estrict ion to s uitable s ubgroups. Recall that up to con- jugation, the elemen tary ab elian su b groups of S 2 n corresp ond to p artit ions of 2 n in to p ow e rs of t w o. As b efore let V n denote the elemen ta ry ab elian subgrou p of S 2 n corresp onding to the trivial partition. In clude S 2 n − 1 × S 2 n − 1 in S 2 n in the standard wa y as in the d eﬁnition of the p rod uct in homology , so that the map on cohomology is th e su mmand ∆ 2 n − 1 , 2 n − 1 of the copro duct ∆. The follo wing Lemma app ears in [17]. Lemma 8.4. The sum of r estriction maps ˆ ρ = ρ V n ⊕ ∆ 2 n − 1 , 2 n − 1 : H ∗ ( B S 2 n ) → H ∗ ( B V n ) ⊕ H ∗ ( B ( S 2 n − 1 × S 2 n − 1 )) is inje ctiv e. Pr o o f. If we consid er all elemen tary ab elian subgroup s of B S 2 n and thus all partitions of 2 n in to p o w ers of tw o , w e see that other th an the trivial partition su c h partitions m ust reﬁn e 2 n = 2 n − 1 + 2 n − 1 . Thus the inclus ions of the corresp onding elemen tary ab elian subgroup s factor up to conjugation through S 2 n − 1 × S 2 n − 1 . The sum of restriction map s ˆ ρ is therefore injectiv e b ecause it factors the r estriction to all elemen tary ab elia n subgrou p s.  THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 25 Pr o o f of The or em 8.3. W e ve rify the equ ality of the th eo rem by v erifying the agreemen t of the restrictions of S q i γ ℓ, 2 k and the sum of outgrowth monomials un der ρ V n and ∆ 2 n − 1 , 2 n − 1 with n = k + ℓ . Corollary 7. 6 states that the restriction of γ ℓ, 2 k to V n is just the Dic kson cla ss d k ,ℓ . In [12 ] Hu’ng calculated the Steenro d squares on Dickso n classes as giv en b y S q i d k ,ℓ =          d k ′ ,ℓ ′ i = 2 k − 2 k ′ d k ′ ,ℓ ′ d k ′′ ,ℓ ′′ i = 2 n + 2 k − 2 k ′ − 2 k ′′ , k ′ ≤ k < k ′′ d k ,ℓ 2 i = 2 k (2 ℓ − 1) 0 o ther wise. By Corollary 7.7 th e restriction to V n is zero for classes whic h are non-trivial transfer pro ducts and sends γ ℓ, 2 k to d k ,ℓ . Th us the outgro wth monomials w e need to consider in our form ula for S q i γ ℓ, 2 k are p r odu cts of on e or tw o γ ℓ ′ , 2 k ′ with ℓ ′ + k ′ = n and (one) ℓ ′ > ℓ . Applying Corollary 7.6 again we see that this agrees with Hu ’ng’s calculatio ns in Theorems A and B of [12]. W e sho w that th e images of S q i γ ℓ, 2 k and the s um of outgrowth m on omials und er ∆ 2 n − 1 , 2 n − 1 agree, b y indu ctio n on k . Since we hav e already v e riﬁed that all restrictions to V n agree then eac h inductive step p ro v es the theorem in that case. If k = 0, the restriction of γ ℓ, 1 is zero. On th e other hand , an outgrowth monomial of S q i γ ℓ, 1 is either zero or a pro duct γ ℓ, 1 γ ℓ − k , 2 k for i = 2 ℓ − 2 k , whic h r estrict s to zero b ecause γ ℓ, 1 do es. In general we ha v e that ∆ 2 n − 1 , 2 n − 1 γ ℓ, 2 k = γ ℓ, 2 k − 1 ⊗ γ ℓ, 2 k − 1 . W e can thus apply the external Cartan form ula to calculate that ∆ 2 n − 1 , 2 n − 1 S q i γ ℓ, 2 k = X p + q = i S q p γ ℓ, 2 k − 1 ⊗ S q q γ ℓ, 2 k − 1 , whic h we und er s ta nd b y ind uctio n to b e the sum of tensor pro ducts of tw o outgro wth monomials for γ ℓ, 2 k − 1 . It thus suﬃces to sho w that the copro duct ∆ 2 n − 1 , 2 n − 1 of the sum of all outg ro wth monomials for γ ℓ, 2 k is the sum of tensor p rodu cts of t w o outgro wth monomials for γ ℓ, 2 k − 1 . This v eriﬁcation is straig h tforw ard using Prop osition 6.6. T hat such copro ducts are giv en b y sums of tensor p rod u cts of t w o su c h monomials is immediate s in ce heigh t and b eing full width are preserve d by copro duct and eﬀectiv e scale can only incr ease on eac h factor. All such tensor pro ducts o ccur sin ce w e can form from m ⊗ m ′ a monomial M where if γ ℓ,p γ ℓ ′ ,p ′ and γ ℓ,q γ ℓ,q ′ are gathered blo c k s in m and m ′ resp ectiv el y then γ ℓ,p + q γ ℓ ′ ,p ′ + q ′ is a gathered b loc k in M . If m and m ′ are outgro wth monomials for γ ℓ, 2 k − 1 then M will b e for γ ℓ, 2 k and m ⊗ m ′ will app ear in its copro duct.  A geometric pro of of this theorem migh t also b e p ossible. Bec ause γ ℓ, 2 k are r epresen te d b y the v arieties Γ ℓ, 2 k , the W u form ula implies that Steenro d op erati ons on them are giv en by Stiefel - Whitney classes of the norm al bund les to these v arieties. Since the Γ ℓ, 2 k are d eﬁned b y equalities of co ord inates, sections can b e p artia lly d eﬁned by p erturb in g those equalities and used for explicit computation. In th e p r evio us s ection we r evisite d and extend ed the classical connection b etw een the cohomol- ogy of symmetric groups an d Dic kson algebras. This connection p ersists when studyin g S teenrod structure. It has long b een kno wn that if ﬁltered appropriately the cohomolo gy of symmetric 26 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA groups has as asso ciated graded a s um of exterior pro ducts of Dic kson algebras, as algebras ov er the S tee nro d algebra. F rom our p oin t of view, we see this using the ﬁltration by num ber of non-trivial transfer p rod ucts and then using the Cartan form ulae w e see that S tee nro d squares do n ot increase ﬁltration. 9. Cup product genera tors after Fe shbach F eshbac h gives in [10] a complete minimal set of ring generators for H ∗ ( B S m ; F 2 ) along with relations which are minimal but not ent irely explicit. Usin g some r esu lts from the pr evious section, we can express h is generators in terms of our Hopf ring generators. The combinato rics of ev en the generating set is somewhat inv olv ed. Deﬁnition 9.1. A lev el- n Dic kson p artiti on of p is an equalit y p = P k 1. The b asic observ ation is that if m is tain ted then q k ( m ) is tain ted for any k . Indeed, u s ing the Cartan formula w e see th at q k ( m ) is s u m of pro ducts of q k i ( b j i ). But using Equation 2, for the j i > 1 the factor q k i ( b j i ) will b e a sum of pro ducts of t w o b ’s of total degree 2 j i + k i ≥ 4, so at least one m ust hav e degree greater than on e. If in q i 1 , ··· i k w e ha v e i k > 1, then b ecause q i k ( b 0 ) = b 0 b i k is tainte d, so w ill b e eve ry te rm in q i 1 , ··· i k ( b 0 ). Thus w i,n m ust ev aluate trivially on an y monomial which is a pro duct of at least one suc h q I . T o see that on th e other hand w i,n do es ev aluate non-trivially on a monomial in the b 1 ,..., 1 , w e calculate q 1 ,..., 1 ( b 0 ). W e get th at q 1 ( b 0 ) = b 0 b 1 , and then that q 1 , 1 ( b 0 ) = q 1 ( q 1 ( b 0 )) = q 1 ( b 1 b 1 ) = q 1 ( b 0 ) q 0 ( b 1 ) + q 0 ( b 0 ) q 1 ( b 1 ) = b 0 b 3 1 + b 3 0 b 3 + b 2 0 b 1 b 2 . In general, q 1 ,..., 1 ( b 0 ) is equal to b 0 b 2 k − 1 1 plus tain ted monomials. Because B ρ ∗ is a map of r in gs, a pro duct of su ch classes in degree i will equal b 0 n − i b 1 i plus tain ted mon omials, and thus b e ev aluated n on-trivia lly b y w i,n , completing the argument.  W e no w d ev elop the com binatorics necessary to expr ess the copro ducts of S tiefel -Whitney classes. Deﬁnition 10.5. A Dic kson bi-partition of the pair ( k , ℓ ) is an equalit y of pairs of p ositiv e in tegers  2 k (2 ℓ − 1) , 2 k + ℓ  = X i  2 k i (2 ℓ i − 1) , 2 k i + ℓ i  . Here w e allo w the tr ivial one-term partition, and we allo w k i to b e zero as we ll as ℓ i to b e zero when th e corresp ondin g k i is. W e manipu lat e su c h a partition as a set p = { ( k i , ℓ i ) } , and sometimes emp hasize the num bers b eing partitioned by writing p = p ( k , ℓ ). W e sa y one bi-partition reﬁn es another if it is obtained by su bstituting of some en try or entries b y corresp ondin g Dic kson bi-partition(s). F or example, b eca use (24 , 32) = (4 , 8) + (6 , 8) + (14 , 16) we ha v e the corresp onding Dic kson bi-parition p (3 , 2) = { (2 , 1) , (1 , 2) , (1 , 3) } . Because in turn of the equalit y (4 , 8) = (0 , 2) + (1 , 2) + (3 , 4), we ha v e that q = { (1 , 0) , (0 , 1) , (1 , 1) , (1 , 2) , (1 , 3 ) } r eﬁnes p . Deﬁnition 10.6. Let Π k ,ℓ denote the set consisting of Dic kson bi-p artitions p expressed as an ordered u nion of t w o smaller partitions p = p ′ ∪ p ′′ , eac h of whic h cont ains no rep eated pairs of n um b ers. Deﬁne a p artial order on Π k ,ℓ b y p ′ ∪ p ′′ ≤ q ′ ∪ q ′′ if p ′ is a (p ossibly trivial) reﬁnement of q ′ and p ′′ of q ′′ . Let φ b e the F 2 -v alued fu nctio n on Π k ,ℓ deﬁned u niquely by X p ′ ∪ p ′′ ≤ q ′ ∪ q ′′ φ ( q ′ ∪ q ′′ ) = 1 , 30 C. GIUSTI, P . SAL V A TO R E, AND D. SINHA for any p ′ ∪ p ′′ ∈ Π k ,ℓ . In other words, the f unction φ is the inv erse under conv olution to the fun cti on wh ic h is one on all of Π k ,ℓ . Thus it could b e determined by M¨ obius inv ersion, though we ha v e not found that to b e enlight ening. Theorem 10.7. A s a Hopf ring, H ∗ ( ` n B S n ) is ge ner ate d by Stiefel-Whitney classes w ( k , ℓ ) . The tr an sfer pr o d uct is exterior, the antip o de is the identity map, and ther e ar e no f urth er r elations other than those given by Hopf ring distributivity and the f act that the pr o duct of classes on diﬀer ent c o mp onents is zer o. The c o pr o duct is given by ∆ w ( k, ℓ ) = X p ′ ∪ p ′′ ∈ Π k,ℓ φ ( p ′ ∪ p ′′ )   K ( k i ,ℓ i ) ∈ p ′ w ( k i , ℓ i )   O   K ( k j ,ℓ j ) ∈ p ′′ w ( k j , ℓ j )   . Pr o o f. That these S tie fel-Whitney classes generate is now an immediate application of Prop osi- tion 10.4 to verify the h yp othesis of Corollary 5.4. Th e lac k of fu rther relations and the additiv e basis f ollo w fr om Theorem 5.3 jus t as this Corollary 5.4 did . The copro duct formula is veriﬁed by d irect c hec k u sing bialgebra structure. By Prop osi- tion 10.4, w ( k , ℓ ) ev aluated on some non-trivial pr o duct m ∗ m ′ whic h is a mon omial will b e non-zero if and only if m and m ′ are pro ducts of ( q 0 ’s and) q 1 ,..., 1 ’s. S uc h pro ducts are in one-to- one corresp ondence with the set Π k ,ℓ . Lo oking at only m , ﬁr st w e express eac h q 1 ,... 1 n uniquely as a pro duct of q 1 ,..., 1 2 k = q 0 ,..... ...., 0 , 1 ,.... , 1 = q ( k , ℓ ), and then r ec ord the ( k , ℓ ) whic h app ear. F or ex- ample, q 0 q 1 3 q 1 , 1 q 1 , 1 , 1 2 = q 0 ∗ q 1 ∗ q 0 , 1 ∗ q 1 , 1 ∗ q 0 , 1 , 1 , 1 corresp onds to { (1 , 0) , (0 , 1) , (1 , 1) , (0 , 2) , (1 , 3) } . Call this b ij ection β from the set of m onomial s in q 1 ,..., 1 to Dickso n bi-partitions. Applying Pr oposition 10.4, w e ﬁ nd that not only do es m ⊗ m ′ pair with J ( k i ,ℓ i ) ∈ β ( m ) w ( k i , ℓ i ) ⊗ J ( k j ,ℓ j ) ∈ β ( m ′ ) w ( k j , ℓ j ) b ut it also p ai rs w ith all similar p rod ucts of Stiefel-Whitney classes o v er q ′ ∪ q ′′ whic h are reﬁned by β ( m ) ∪ β ( m ′ ). Th us, if w e tak e the linear combinatio n with co eﬃcien ts giv en by φ , that sum will p air to one w ith m ⊗ m ′ .  Referen ces 1. John F rank A dams, Inﬁni te l o op sp ac es , An nals of Mathematics Stud ies, vol. 90, Princeton U n iv ersit y Press, Princeton, N.J., 1978. MR 505692 (80d:55001) 2. Alejandro Adem, John Maginnis, and R. James Milgram, Symmetric invariants and c ohomolo gy of gr oups , Math. Ann. 287 (1990), no. 3, 391–411. MR 1060683 ( 91 i:55022 ) 3. Alejandro A dem and R. James Milgram, Cohomolo gy of ﬁnite gr oups , Grun dlehren der Mathematisc hen Wissensc haften [F un damen tal Principles of Mathematical Sciences], vol. 309, S p ringer-V erlag, Berlin, 1994. MR MR13170 96 (96f:20082) 4. Jos ´ e A dem, The r elations on Ste enr o d p owers of c ohomolo g y classes. Algeb r a ic ge o metry and top olo gy , A symp o- sium in honor of S. Lefschetz, Princeton Universit y Press, Princeton, N. J., 1957, p p . 191–238. MR MR0085502 (19,50c) 5. M. F. Atiyah and G. B. Segal, Equivariant K -the or y and c ompletion , J. Diﬀerential Geometry 3 (1969), 1–18. MR MR02599 46 (41 #4575) 6. Mic hael Barratt and Stewar t Priddy , On the homolo gy of non-c onne cte d monoids and their asso ciate d gr o ups , Commen t. Math. Helv. 47 (1972), 1–14. MR MR0314940 (47 #3489) 7. T errence P . Bisson and And r ´ e Jo yal , Q -rings and the homolo gy of the symmetric gr oups , O perads: Pro ceed- ings of R enaissa nce Conferences (Hartford, CT/Lumin y , 1995), Contemp. Math., vo l. 202, Amer. Math. So c., Pro vidence, RI, 1997, pp. 235–286 . MR MR1436923 (98e:55021) THE M OD-TW O COHOMO LOGY RINGS OF SYMMETRIC GR OUPS 31 8. R. R. Bru n er, J. P . Ma y , J. E. McClure, and M. Steinberger, H ∞ ring sp e ctr a and their applic ations , Lecture Notes in Mathematics, vo l. 1176, Springer-V erlag, Berlin, 1986. MR MR836132 (88e:55001) 9. F rederick R. Cohen, Thomas J. Lada, and J. Peter Ma y , The homolo gy of iter ate d lo op sp ac es , Springer-V erlag, Berlin, 1976. MR MR0436146 (55 #9096) 10. Mark F eshbac h, The mo d 2 c ohomolo gy rings of the symmetric gr oups and invariants , T op olo gy 41 (2002), no. 1, 57–84 . MR MR1871241 (2002h:20074) 11. J. H. Gunaw ardena, J. Lannes, and S . Zarati, Cohomolo gie des gr oup es sym´ etriques et applic ation de Quil l en , Adv ances in homotopy theory (Cortona, 1988), London Math. So c. Lecture N ote Ser., vol. 139, Cambridge Univ. Press, Cam bridge, 1989, pp. 61–68. MR 1055868 (91d:18013) 12. Nguyˆ en H. V. Hu ng, The action of the Ste enr o d squar es on the mo dular invariants of line ar gr oups , Pro c. Amer. Math. Soc. 113 (1991), no. 4, 1097–1104. MR 1064904 (92c:55018) 13. Nguyen H ˜ u’u V iet Hu ’ng, The mo dulo 2 c ohomolo gy algebr as of symmetric gr oups , Japan. J. Math. (N .S.) 13 (1987), no. 1, 169–2 08. MR MR914318 (89g:55028) 14. Donald Knutson, λ -rings and the r epr esentation the ory of the symmetric gr oup , Lecture Notes in Mathematics, V ol. 308, S pringer-V erlag, Berlin, 1973. MR MR0364425 (51 #679) 15. Stanley O. Kochman, Homol o gy of the classic al gr oups over the Dyer-Lashof algebr a , T rans. Amer. Math. Soc. 185 (1973), 83–136 . MR MR0331386 (48 #9719) 16. I. G. Macdonald, Symmetric functions and Hal l p olynomials , second ed., Oxford Mathematical Monographs, The Clarendon Press Ox fo rd Universit y Press, N ew Y ork, 1995, With contributions by A. Zelevinsk y , Oxford Science Pu blica tions. MR 1354144 (96h:05207) 17. Ib Madsen and R. James Milgram, The classifying sp ac es for sur ge ry and c o b or dism of manifol ds , A nnals of Mathematics Studies, vol . 92, Princeton Universit y Press, Princeton, N.J., 1979. MR MR548575 (81b:57014) 18. R. James Milgram, Private c ommunic ation. 19. , The mo d 2 spheric al char act eristic classes , Ann . of Math. (2) 92 (1970), 238–261. MR MR0263100 (41 #7705) 20. John W . Milnor and John C. Mo ore, On the structur e of Hopf al geb r as , Ann. of Math. (2) 81 (1965), 211–264 . MR MR01740 52 (30 #4259) 21. Minoru Nak ao k a, Homolo gy of the inﬁnite symmetric gr oup , A nn. of Math. (2) 73 (1961), 229–257. MR MR01318 74 (24 #A1721) 22. Stewa rt Priddy , Dyer-Lashof op er ations for the classifying sp ac es of c ert ain matrix gr ou ps , Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 102, 179–19 3. MR MR0375309 (51 #11505) 23. D. G. Qu ill en, The sp e ctrum of an e quivariant c ohomolo gy ring, i. , Ann. of Math. (2) 94 (1971), 549–5 72. 24. D. G. Qu ill en and B. B. V enkov, Cohomolo gy of ﬁnite gr oups and elementary ab el ian sub gr oups. , T op ology 11 (1972), 317–318 . 25. Douglas C. Rav enel and W. Stephen Wilson, The Hopf ring for c om plex c ob or dism , J. Pure A ppl. Algebra 9 (1976/77 ), no. 3, 241–280. MR MR0448337 (56 #6644) 26. N. E. St eenrod, Homolo gy gr oups of symmetric gr oups and r e duc e d p ower op er ations , Pro c. Nat. A cad. Sci. U. S. A . 39 (1953), 213–217. MR MR0054964 (14,1005d) 27. N. P . Strickland, Mor ava E -the ory of symmetric gr oups , T op ology 37 (1998), no. 4, 757–77 9. MR MR16077 36 (99e:5500 8) 28. Neil P . Strickland and P aul R. T u rn er, R ational Mor ava E -the ory and D S 0 , T op ology 36 (1997), no. 1, 137–151. MR MR14104 68 (97g:55005) 29. Andrey V. Zelevinsky , R epr esentations of ﬁnite classic al gr oups , Lecture Notes in Mathematics, vol. 869, Springer-V erlag, Berlin, 1981, A Hopf algebra approac h. MR MR643482 ( 83k:20017) Ma the ma tics Dep ar tment, Willamette Universi ty E-mail addr ess : cgiusti@willame tte.edu Dip ar timento di Ma tema tica, Uni v ersit ` a di Roma Tor Ve r ga t a E-mail addr ess : salvator@mat.un iroma2.it Ma the ma tics Dep ar tment, Univer si ty of Oregon E-mail addr ess : dps@math.uorego n.edu

Mod-two cohomology of symmetric groups as a Hopf ring

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