Multispecies inhomogeneous $t$-PushTASEP with general capacity

MUL TISPECIES INHOMOGENEOUS t -PUSHT ASEP WITH GENERAL CAP A CITY AR VIND A YYER AND A TSUO KUNIBA Abstract. W e study an n -species t -PushT ASEP , an integrable long-range sto chastic process, on a one- dimensional perio dic lattice with inhomogeneities x 1 , . . . , x L and arbitrary capacity l at eac h lattice site. The Mark ov matrix is iden tiﬁed with an alternating sum of commuting transfer matrices ov er all fundamen- tal representations of U t ( b sl n +1 ). Stationary probabilities are expressed in a matrix pro duct form involv- ing a fusion of quantized corner transfer matrices for the strange ﬁve-v ertex mo del in tro duced b y Ok ado, Scrimshaw, and the second author. The resulting partition function, which serv es as the normalization factor of the stationary probabilities, is obtained from the l = 1 case by a ﬁnite pleth ystic substitution of length l . 1. Introduction PushT ASEP is a class of totally asymmetric simple exclusion pro cesses (T ASEPs) of in teracting particles on one-dimensional lattice. Its characteristic feature is a sto c hastic dynamics in whic h multiple particles ma y mov e simultaneously ov er long distances b y pushing one another according to prescrib ed rules. By no w, v arious versions of PushT ASEPs hav e b een in troduced and studied extensively; see, for example, [ ANP23 , AK25 , AM23 , AMW24 , BW22 , CP13 , P19 ] and the references therein. In this pap er w e study a PushT ASEP form ulated as a con tinuous-time Mark o v process on a perio dic lattice of length L , in whic h eac h local state is given b y a length- l ro w-shap ed semistandard tableau with en tries in { 0 , 1 , . . . , n } , namely , { ( i 1 , . . . , i l ) ∈ Z l | 0 ≤ i 1 ≤ · · · ≤ i l ≤ n } . Suc h a lo cal state may b e in terpreted as an assem bly of particles of sp ecies 1 , 2 , . . . , n , together with δ i 1 , 0 + · · · + δ i l , 0 v acant slots, within the maximal capacit y l at each site. (In the main text, ho wev er, we treat 0 as a particle sp ecies as well.) The transition rates dep end on the parameter t and on inhomogeneity parameters x 1 , . . . , x L attac hed to eac h site. The mo del is Mark o vian, and hence physically meaningful, in the parameter region t ≥ 0 and x 1 , . . . , x L > 0. W e refer to this model, which has  n + l l  lo cal states at eac h site, as the n -sp ecies capacit y- l t -PushT ASEP . A k ey ingredien t underlying our approach is the quantum R -matrix S k l ( z ) acting on V k ⊗ V l in an ap- propriate gauge, where V k and V l denote the degree- k antisymmetric tensor representation and the degree- l symmetric tensor represen tation of the quantum aﬃne algebra U t ( b sl n +1 ), resp ectiv ely . This coupling of anti- symmetric and symmetric tensor representations appears to b e new in the con text of in tegrable probabilit y . In principle, S k l ( z ) can b e obtained [ KRS81 ] either b y l -fold symmetric fusion of S k 1 ( z ) or by k -fold anti- symmetric fusion of S 1 l ( z ). In this paper, w e adopt a more eﬃcient construction based on three-dimensional in tegrability explored in [ K22 , Chap. 11.3]. It directly yields an explicit form ula for S k l ( z ) given in ( 52a )– ( 53b ) in terms of the 3D L -operator ( 50 ). Based on the R -matrix, we construct the commuting family of transfer matrices T k ( z | x 1 , . . . , x L ) of the solv able vertex mo del (cf. [ Bax83 ]) with sp ectral parameter z and inhomogeneities x 1 , . . . , x L for 0 ≤ k ≤ n +1. In the terminoogy of the quan tum inv erse scattering method [ STF80 ], it has the auxiliary space V k and acts on the quan tum space V ⊗ L l . Let H n,l ( x 1 , . . . , x L ) be the Marko v matrix of the n -sp ecies capacity- l t -PushT ASEP in ( 17a ) whose tran- sition rates are given b y ( 29 ) and also describ ed com binatorially in Section 2.3 . Our ﬁrst main result is the follo wing form ula (Theorem 11 ): H n,l ( x 1 , . . . , x L ) = D − 1 m n +1 X k =0 ( − 1) k − 1 dT k ( z | x 1 , . . . , x L ) dz      z =0 −  L X j =1 1 x j  Id . (1) Date : F ebruary 20, 2026. 2020 Mathematics Subje ct Classiﬁc ation. 60J27, 82B20, 82B23, 82B44, 81R50, 17B37. Key words and phrases. t -PushT ASEP , multispecies, inhomogeneous, Y ang–Baxter equation. 1 2 AR VIND A YYER AND A TSUO KUNIBA Here, D m is a scalar factor deﬁned in ( 15 ), determined by the particle multiplicit y m . The RHS is an alternating sum of the deriv ative of the transfer matrices whose auxiliary spaces range ov er all fundamen tal represen tations V 0 , . . . , V n +1 . The iden tity ( 1 ) extends a Baxter-type form ula for quan tum Hamiltonians (cf. [ Bax83 , eq. (10.14.20)]) to an inhomogeneous sto chastic setting, generalizing our earlier result for the l = 1 case [ AK25 ] to arbitrary capacity . A notable asp ect of ( 1 ) is that neither the individual transfer matrix T k ( z | x 1 , . . . , x L ) nor its deriv ative at z = 0 is sto c hastic in general: their matrix elements need not b e p ositiv e and do not satisfy probability conserv ation. Nev ertheless, the alternating sum in ( 1 ) acts as an inclusion–exclusion mechanism, retaining admissible particle motions with correct rates while cancelling forbidden c hannels. The result ( 1 ) reduces the problem of ﬁnding stationary states of the mo del to that of constructing a join t eigenstate of the commuting transfer matrices. Our second main result provides suc h a construction explicitly: the stationary probability of a conﬁguration ( σ 1 , . . . , σ L ) is expressed in a matrix product form P ( σ 1 , . . . , σ L ) = T r ( A σ 1 ( x 1 ) · · · A σ L ( x L )) , (2) up to normalization. Here the op erator A i ( z ) asso ciated with a lo cal state i = ( i 1 , . . . , i l ) is deﬁned as (cf. ( 120 ), ( 123 )) A i ( z ) = X A i ′ 1 ( z ) A i ′ 2 ( tz ) · · · A i ′ l ( t l − 1 z ) , (3) where the sum runs ov er distinct p erm utations ( i ′ 1 , . . . , i ′ l ) of ( i 1 , . . . , i l ). F or the basic case l = 1, the op erators A 0 ( z ) , . . . , A n ( z ) in ( 113 ) coincide, up to a minor con v en tional c hange, with those in tro duced in [ K OS24 ]. They are c orner tr ansfer matric es (CTMs [ Bax83 , Chap.13]) of the strange ﬁve-v ertex model, whic h are quantize d in the sense that the “Boltzmann weigh ts” take v alues in the t -oscillator algebra ( 109 ). The construction of the CTMs for higher l in ( 3 ) ma y b e viewed as a symmetric fusion at the lev el of matrix pro duct operators. Let Z l, m ( x 1 , . . . , x L ; t ) = P ( σ 1 ,..., σ L ) ∈S ( m ) P ( σ 1 , . . . , σ L ) denote the normalization factor of the stationary probabilities for capacity l , where S ( m ) is deﬁned in ( 13 ). This quan tit y deﬁnes a symmetric polynomial in x 1 , . . . , x L and is commonly referred to as the partition function. F rom ( 2 ) and ( 3 ), it admits a simple reduction to the l = 1 case: Z l, m ( x 1 , . . . , x L ; t ) = Z 1 , m  1 − t l 1 − t x 1 , . . . , 1 − t l 1 − t x L ; t  , (4) where the notation (1 − t l ) x j / (1 − t ) represents the length- l plethystic substitution x j 7→ x j , tx j , . . . , t l − 1 x j for each j . Consequently , the righ t-hand side inv olv es l L inhomogeneit y parameters geometrically weigh ted as x 1 , tx 1 , . . . , t l − 1 x 1 , . . . , x L , tx L , . . . , t l − 1 x L , whic h corresp ond to a system with l L sites. F or l = 1, the partition function Z 1 , m ( x 1 , . . . , x L ; t ) has essentially b een identiﬁed with a Macdonald p olynomial at q = 1 [ AMW24 , CMW22 ]. (See also [ CDW15 ] for an earlier result in the con text of ASEP .) More generally , symmetric fusion of R -matrices, or of the asso ciated vertex models, has been observ ed to manifest itself as a plethystic substitution at the lev el of partition functions [ GW20 , M25 ]. The relation ( 4 ) giv es an explicit realization of this phenomenon for general capacit y l . Our mo del reduces to the one studied in [ AMW24 ] in the special case l = 1. F or general capacity l , the inhomogeneous n -sp ecies t -PushT ASEP has also b een studied in [ BW22 , Sec. 12.5] and [ ANP23 , Sec. 6]. Although the formulations adopted in those works app ear rather diﬀerent from the one presented here, the underlying dynamics is exp ected to commute with the Marko v and transfer matrices constructed in this pap er under perio dic b oundary conditions, since b oth approaches are based on R -matrices for U t ( b sl n +1 ) and act on the same state space. The present pap er fo cuses on the explicit realization of the Marko v matrix as an alternating sum of transfer matrices and on the matrix pro duct formula for the stationary probabilities, asp ects that ha ve not been addressed so far. The outline of the pap er is as follows. In Section 2 , w e deﬁne the n -sp ecies capacity- l t -PushT ASEP and pro vide a combinatorial description of its transition rates. In Section 3 , w e present the R -matrix S k l ( z ) based on the three-dimensional approach [ K22 , Chap.11], together with an appropriate gauge choice. Its matrix elemen ts are describ ed in detail, which will b e used in Section 5 . In Section 4 , we give a standard construction of the commuting transfer matrices T k ( z | x 1 , . . . , x L ) from the R -matrix S k l ( z ). In Section 5 , w e prov e Theorem 11 , whic h identiﬁes the Marko v matrix in troduced in Section 2 with an alternating sum of the deriv atives of the transfer matrices ev aluated at z = 0. The strategy of the proof parallels that of the l = 1 case [ AK25 ]. In Section 6 , we introduce the quan tized corner transfer matrices A i ( z ) and explain their MUL TISPECIES t -PUSHT ASEP 3 prop erties, most notably the Zamolo dc hik o v–F addeev algebra. In Section 7 , we derive the matrix pro duct form ula for the stationary probabilities, building on the results of the preceding sections. W e also present a few immediate consequences for the partition function. App endix A recalls the quantum R -matrix S k,l ( z ) on symmetric tensor represen tations in a gauge adapted to the presen t paper. 2. n -species t -PushT ASEP with cap acity l 2.1. Preliminary. Given n ≥ 1, deﬁne the following v ector spaces and index sets: V k = M i ∈ B k C v i , B k = { i = ( i 0 , . . . , i n ) ∈ { 0 , 1 } n +1 | | i | = k } (0 ≤ k ≤ n + 1) , (5) V l = M i ∈ B l C v i , B l = { i = ( i 0 , . . . , i n ) ∈ ( Z ≥ 0 ) n +1 | | i | = l } ( l ∈ Z ≥ 0 ) , (6) | i | = i 0 + · · · + i n , e j = ( δ j, 0 , . . . , δ j,n ) ∈ B 1 = B 1 (0 ≤ j ≤ n ) . (7) The lab el i = ( i 0 , . . . , i n ) of a basis vector is referred to as the multiplicity r epr esentation . W e further introduce the following sets of integer arrays: T k = { ( I 1 , . . . , I k ) ∈ Z k | 0 ≤ I 1 < · · · < I k ≤ n } (0 ≤ k ≤ n + 1) , (8) T l = { ( I 1 , . . . , I l ) ∈ Z l | 0 ≤ I 1 ≤ · · · ≤ I l ≤ n } ( l ∈ Z ≥ 0 ) . (9) Elemen ts of T k (resp. T l ) are in terpreted as semistandard tableaux of column shap e with depth k (resp. ro w shap e of width l ), ﬁlled with entries from { 0 , 1 , . . . , n } . W e identify T k with B k (resp. T l with B l ) through the bijection i α = δ α,I 1 + · · · + δ α,I k (0 ≤ α ≤ n ) . (10) W e refer to ( 8 ) and ( 9 ) as the table au r epr esentation of the basis of V k and V l , resp ectiv ely . Both represen tations, the multiplicit y representation and the tableau representation, will b e used throughout this pap er. The tableau representation aligns naturally with a particle interpretation, whereas many form ulas are more conv enien tly expressed in the multiplicit y representation. F or tw o elements σ , σ ′ ∈ B l , we deﬁne the relation σ ≺ σ ′ def ⇐ ⇒ σ − σ ′ = e r 1 − e r 2 + · · · + e r µ − 1 − e r µ for some 0 ≤ r 1 < · · · < r µ ≤ n and µ ∈ 2 Z ≥ 1 . (11) The RHS represen ts the alternating sum P µ i =1 ( − 1) i − 1 e r i . W e write σ ⪯ σ ′ if σ ≺ σ ′ or σ = σ ′ . Note that σ ⪯ σ ′ and σ ′ ⪯ σ ′′ do not imply σ ⪯ σ ′′ . W e use the notation θ (true) = 1 and θ (false) = 0 throughout. 2.2. Deﬁnition of n -sp ecies capacit y- l inhomogeneous t -PushT ASEP. F or n, l ≥ 1, w e introduce the n -sp ecies, capacity- l , inhomogeneous t -PushT ASEP on a one-dimensional p eriodic lattice of length L . It is a contin uous-time Marko v pro cess on the space V = V ⊗ L l , where V l is deﬁned in ( 6 ). W e often write a basis v ector v σ 1 ⊗ · · · ⊗ v σ L simply as | σ 1 , . . . , σ L ⟩ or |  σ ⟩ , with the shorthand  σ = ( σ 1 , . . . , σ L ) with σ j = ( σ j, 0 , . . . , σ j,n ) ∈ B l . A vector v σ ∈ V l , or equiv alen tly the lab el σ ∈ B l , is regarded as a lo c al state . In the multiplicity r epr esentation , a lo cal state σ = ( σ 0 , . . . , σ n ) ∈ B l means that the site con tains σ α particles of sp ecies α for 0 ≤ α ≤ n . 1 On the other hand, the table au r epr esentation σ = ( I 1 , . . . , I l ) ∈ T l lists the sp ecies of the l particles present at the site. The integer parameter l is referred to as the c ap acity , meaning the maximum num ber of particles that can o ccup y a single site. Let V ( m ) ⊂ V b e the subspace sp eciﬁed b y the multiplicity m = ( m 0 , . . . , m n ) ∈ ( Z ≥ 1 ) n +1 of the particles as follows: V ( m ) = M ( σ 1 ,..., σ L ) ∈S ( m ) C | σ 1 , . . . , σ L ⟩ , (12) S ( m ) = { ( σ 1 , . . . , σ L ) | σ i = ( σ i, 0 , . . . , σ i,n ) ∈ B l , σ 1 ,α + · · · + σ L,α = m α (0 ≤ α ≤ n ) } . (13) 1 W e regard 0 also as a particle, but follow the con v entional terminology and refer to the mo del as an n -species system. 4 AR VIND A YYER AND A TSUO KUNIBA Note that m 0 + · · · + m n = Ll . W e set K α = ( l for α = 0 , m 0 + · · · + m α − 1 for 1 ≤ α ≤ n, K α = m 0 + · · · + m α − 1 for 0 ≤ α ≤ n, (14) D m = n Y α =0 (1 − t K α ) . (15) W e shall exclusively consider the case m 0 , . . . , m n ≥ 1 throughout the article, hence K 0 , . . . , K n ≥ 1, K 0 = 0, K 1 , . . . , K n ≥ 1 and D m  = 0. The n -sp ecies capacity- l inhomogeneous t -PushT ASEP is a sto chastic pro cess on each V ( m ) gov erned by the master equation d ds | P ( s ) ⟩ = H n,l ( x 1 , . . . , x L ) | P ( s ) ⟩ , (16) where the state vector is giv en by | P ( s ) ⟩ = P ( σ 1 ,..., σ L ) ∈S ( m ) P ( σ 1 , . . . , σ L ; s ) | σ 1 , . . . , σ L ⟩ , and P ( σ 1 , . . . , σ L ; s ) denotes the probability that the conﬁguration ( σ 1 , . . . , σ L ) o ccurs at time s . The Marko v matrix H n,l = H n,l ( x 1 , . . . , x L ) : V ( m ) → V ( m ) is given by H n,l |  σ ⟩ = X  σ ′ ∈S ( m )  σ ′  =  σ L X o =1 1 x o Y 0 ≤ h ≤ n w ( o )  σ ,  σ ′ ( h ) |  σ ′ ⟩ +  L X o =1 C  σ ,o ( t ) − 1 x o  |  σ ⟩ , (17a) C  σ ,o ( t ) = Y 0 ≤ h ≤ n 1 − t K h + σ o,h 1 − t K h , (17b) where  σ = ( σ 1 , . . . , σ L ) and  σ ′ = ( σ ′ 1 , . . . , σ ′ L ) with σ j = ( σ j, 0 , . . . , σ j,n ) , σ ′ j = ( σ ′ j, 0 , . . . , σ ′ j,n ) ∈ B l . Each parameter x o > 0 asso ciated with a lattice site o ∈ { 1 , . . . , L } represents the site-wise inhomogeneity of the system. The factor w ( o )  σ ,  σ ′ ( h ) is a rational function of t , and constitutes the main part of H n,l . Its explicit deﬁnition will b e giv en b elo w. Let  σ = ( σ 1 , . . . , σ L ) and  σ ′ = ( σ ′ 1 , . . . , σ ′ L ) ∈ S ( m ) with σ j = ( σ j, 0 , . . . , σ j,n ) , σ ′ j = ( σ ′ j, 0 , . . . , σ ′ j,n ) ∈ B l . A necessary condition 2 for w ( o )  σ ,  σ ′ ( h )  = 0 for  σ  =  σ ′ is that  σ ′ is obtained from  σ by a sequence of push-out mo ves inv olving particles h g , h g − 1 , . . . , h 1 , h 0 in this order, for some 0 ≤ h 0 < · · · < h g ≤ n and 1 ≤ g ≤ n . 3 particles h g h g − 1 · · · h 1 h 0 departure site o = p ( h g ) p ( h g − 1 ) · · · p ( h 1 ) p ( h 0 ) arriv al site p ′ ( h g ) p ′ ( h g − 1 ) · · · p ′ ( h 1 ) p ′ ( h 0 ) = o T able 1. List of mo ving particles. The relations p ′ ( h α ) = p ( h α − 1 ) (1 ≤ α ≤ g ) and p ′ (0) = p ( h g ) = o are assumed. T able 1 illustrates a pro cess in whic h a particle of species h g departs from site o and arriv es at site p ′ ( h g ) = p ( h g − 1 ), thereby pushing out a smaller particle of sp ecies h g − 1 . The particle h g − 1 in turn mov es to p ′ ( h g − 1 ) = p ( h g − 2 ), pushing out h g − 2 , and this cascading motion contin ues until a particle h 0 ﬁnally reaches and r eﬁl ls the original site o . This reﬁll is necessary b ecause we regard 0 as a particle sp ecies as w ell, and eac h site is required to accommo date exactly l particles. The departure and arriv al sites m ust satisfy the relations p ′ ( h α ) = p ( h α − 1 ) (1 ≤ α ≤ g ) , p ′ ( h 0 ) = p ( h g ) = o, (18) σ p ( h α ) ,h α ≥ 1 , p ( h α )  = p ′ ( h α ) (0 ≤ α ≤ g ) . (19) These conditions do not exclude the p ossibilit y p ( h α ) = p ′ ( h β ) for α > β ≥ 0. In particular, a single site ma y exp erience multiple (up to l ) push-out even ts during a pro cess. See Figure 1 . 2 Suﬃciency requires an additional condition σ ′ o, 0 ≥ 1, as will be shown in Proposition 1 . 3 F or simplicity , we refer to a “particle of sp ecies h ” simply as “particle h ”. Species 0 is also regarded as a particle for the purpose of this description. MUL TISPECIES t -PUSHT ASEP 5 − r 4 + s 6 − s 5 + u 4 − u 3 + r 3 − r 2 + v 4 − v 3 + s 4 − s 3 + t 2 − t 1 + s 2 − s 1 + u 2 − u 1 + v 2 − v 1 + r 1 h 9 h 8 h 7 h 7 h 6 h 5 h 5 h 4 h 3 h 3 h 2 h 1 h 0 h 0 · · · · · · · · · · · · · · · · · · · · · · · · n 0 p ( h 9 ) = p ′ ( h 7 ) = p ( h 6 ) = p ′ ( h 0 ) = o p ′ ( h 9 ) = p ( h 8 ) = p ′ ( h 5 ) = p ( h 4 ) = p ′ ( h 3 ) = p ( h 2 ) = j 1 p ′ ( h 4 ) = p ( h 3 ) = j 2 p ′ ( h 8 ) = p ( h 7 ) = p ′ ( h 2 ) = p ( h 1 ) = j 3 p ′ ( h 6 ) = p ( h 5 ) = p ′ ( h 1 ) = p ( h 0 ) = j 4 Figure 1. Schematic plot of a transition  σ →  σ ′ in T able 1 for the case 0 ≤ h 0 < · · · < h g =9 ≤ n . The vertical axis ranges from 0 to n , corresp onding to the comp onen ts of arrays in B l deﬁned in ( 6 ), while the lattice sites 1 , . . . , L are aligned along the horizon tal axis with p eriodic b oundary conditions. At each p oin t ( j, h ) ∈ { 1 , . . . , L } × [0 , n ], the symbol ⊕ h , ⊖ h , or blank is placed according to σ ′ j,h − σ j,h = 1, − 1, or 0, respectively , where  σ = ( σ 1 , . . . , σ L ) with σ j = ( σ j, 0 , . . . , σ j,n ) ∈ B l , and  σ ′ = ( σ ′ 1 , . . . , σ ′ L ) with σ ′ j = ( σ ′ j, 0 , . . . , σ ′ j,n ) ∈ B l in the multiplicit y representation. Only the sites where the lo cal state changes are depicted and lab eled by o, j 1 , . . . , j 4 ; all other sites are omitted in accordance with the notion of a r e duc e d diagr am in tro duced in Section 5.2 . F or instance, σ ′ o − σ o = e r 1 − e r 2 + e r 3 − e r 4 for some 0 ≤ r 1 < · · · < r 4 ≤ n , consistent with ( 25 ), and σ ′ j 2 − σ j 2 = − e t 1 + e t 2 for some 0 ≤ t 1 < t 2 ≤ n , as in ( 27 ). A segment of the form ⊖ p h q ⊕ p ′ is understo o d to assume h q = p = p ′ , and represents the mov emen t of a particle of sp ecies h q from the site with ⊖ p to that with ⊕ p ′ , under the p eriodic b oundary condition. In column o , a vertical line en tering the diagram from the top and an arro w exiting down ward are added, so that it ma y b e viewed as a path descending through the system, p ossibly wrapping around it. This viewp oin t will b e useful for the combinatorial description of the transition rates in Section 2.3 . 6 AR VIND A YYER AND A TSUO KUNIBA Let { ¯ h 1 , . . . , ¯ h n − g } = { 0 , . . . , n } \ { h 0 , . . . , h g } ; that is, { 0 , . . . , n } = { h 0 , . . . , h g } ⊔ { ¯ h 1 , . . . , ¯ h n − g } . (20) All particles of sp ecies ¯ h 1 , . . . , ¯ h n − g remain within their original sites. The ab ov e condition w ( o )  σ ,  σ ′ ( h )  = 0 can b e equiv alently stated as follows: (i) o is the unique site such that σ ′ o ≺ σ o in the sense of ( 11 ). F or all other sites j  = o , one has σ j ⪯ σ ′ j . (ii) Let σ j = ( σ j, 0 , . . . , σ j,n ) and σ ′ j = ( σ ′ j, 0 , . . . , σ ′ j,n ). (ii-i) F or each h ∈ { h 0 , . . . , h g } , there exists exactly one site p ( h ) such that σ ′ p ( h ) ,h = σ p ( h ) ,h − 1, and exactly one site p ′ ( h ) such that σ ′ p ′ ( h ) ,h = σ p ′ ( h ) ,h + 1. (ii-ii) F or each h ∈ { ¯ h 1 , . . . , ¯ h n − g } , one has σ j,h = σ ′ j,h for all 1 ≤ j ≤ L . W e deﬁne the factor w ( o )  σ ,  σ ′ ( h ) in ( 17a ) as follows: w ( o )  σ ,  σ ′ ( h ) =        (1 − t σ p ( h ) ,h ) t ℓ h 1 − t K h if h ∈ { h 0 , . . . , h g } , 1 − t K h +Φ h 1 − t K h if h ∈ { ¯ h 1 , . . . , ¯ h n − g } . (21) Here, ℓ h is deﬁned by ℓ h = n umber of particles in  σ with sp ecies in [0 , h ) in the cyclic (clo ckwise) interv al [ p ( h ) , p ′ ( h )) including p ( h ) but excluding p ′ ( h ) . (22) F or l = 1, this reduces to the num ber of particles with sp ecies in [0 , h ) within the cyclic (clo c kwise) interv al ( p ( h ) , p ′ ( h )), excluding b oth endp oin ts, in agreement with ℓ h in [ AK25 , eq.(2.7)]. The quantit y Φ h = Φ h (  σ ,  σ ′ ) is given by Φ h = φ ′ h ( σ o , σ ′ o ) + X 1 ≤ j ≤ L j  = o φ h ( σ j , σ ′ j ) . (23) T o deﬁne φ ′ h ( σ o , σ ′ o ), we use the assumption σ ′ o ≺ σ o in (i) and recall the deﬁnition ( 11 ). Then: φ ′ h ( σ o , σ ′ o ) = ( σ o,h if h ∈ [0 , r 1 ) ⊔ ( r 2 , r 3 ) ⊔ · · · ⊔ ( r µ − 2 , r µ − 1 ) ⊔ ( r µ , n ] , 0 otherwise , (24) where the num bers 0 ≤ r 1 < r 2 < · · · < r µ ≤ n (with µ ∈ 2 Z ≥ 1 ) are deﬁned by σ ′ o − σ o = e r 1 − e r 2 + · · · + e r µ − 1 − e r µ . (25) The function φ h ( σ j , σ ′ j ) should b e deﬁned for j  = o and σ j ⪯ σ ′ j in the light of the condition in (i). W e set φ h ( σ j , σ ′ j ) = ( σ j,h if σ j ≺ σ ′ j and h ∈ ( r 1 , r 2 ) ⊔ ( r 3 , r 4 ) ⊔ · · · ⊔ ( r µ − 1 , r µ ) , 0 otherwise , (26) where, for σ j ≺ σ ′ j , the sequence 0 ≤ r 1 < r 2 < · · · < r µ ≤ n (with µ ∈ 2 Z ≥ 1 ) is deﬁned via σ j − σ ′ j = e r 1 − e r 2 + · · · + e r µ − 1 − e r µ . (27) Observ e that φ 0 ( σ j , σ ′ j ) = 0 for all j  = o , since 0 ∈ ( r 1 , r 2 ) ⊔ ( r 3 , r 4 ) ⊔ · · · ⊔ ( r µ − 1 , r µ ) in ( 26 ) for an y sequence 0 ≤ r 1 < · · · < r µ ≤ n . Consequently , when h = 0, the deﬁnition( 23 ) simpliﬁes to Φ 0 = φ ′ 0 ( σ o , σ ′ o ) . (28) F rom ( 21 ), the main factor in the transition rate in ( 17a ) b ecomes n Y h =0 w ( o )  σ ,  σ ′ ( h ) = Y h ∈{ h 0 ,...,h g } (1 − t σ p ( h ) ,h ) t ℓ h 1 − t K h Y h ∈{ ¯ h 1 ,..., ¯ h n − g } 1 − t K h +Φ h 1 − t K h . (29) Prop osition 1. In the pr o c ess  σ →  σ ′ describ e d in T able 1 , the pr o duct ( 29 ) with generic t vanishes if and only if the fol lowing c ondition is satisﬁe d: σ ′ o, 0 = 0 . (30) MUL TISPECIES t -PUSHT ASEP 7 Pr o of. Note that σ p ( h ) ,h ≥ 1 by ( 19 ), K 0 , . . . , K n ≥ 1 by ( 14 ), and Φ 0 , . . . , Φ n ≥ 0 by ( 23 )–( 26 ). Therefore, the only factor in ( 29 ) that can v anish is (1 − t Φ 0 ), which o ccurs when 0 ∈ { ¯ h 1 , . . . , ¯ h n − g } . Supp ose 0 ∈ { ¯ h 1 , . . . , ¯ h n − g } , i.e., particles of species 0 do not mov e. This implies r 1 ≥ 1 in ( 25 ), hence 0 ∈ [0 , r 1 ) in ( 24 ). Therefore, we obtain Φ 0 ( 28 ) = φ ′ 0 ( σ o , σ ′ o ) = σ o, 0 = σ ′ o, 0 . Th us, if ( 29 ) v anishes, it must b e that σ ′ o, 0 = 0, proving the “only if” part. Con versely , supp ose ( 30 ) holds. Then σ o, 0 = 0 must also hold, since otherwise σ o, 0 ≥ 1 would con tradict ( 25 ). The fact that σ ′ o, 0 = σ o, 0 = 0 implies that sp ecies 0 particles either do not mo ve at all, or , if they do, their motion must b e conﬁned to sites other than o . Can there exist such a particle of sp ecies 0 that mov es from a site α to a diﬀerent site β within { 1 , 2 , . . . , L } \ { o } ? The answ er is no, since such a mo v e w ould imply σ β , 0 < σ ′ β , 0 , which contradicts the inequality σ β ≺ σ ′ β in condition (i) of the assumption (see ( 11 )). It follo ws that sp ecies 0 particles do not mov e, so 0 ∈ { ¯ h 1 , . . . , ¯ h n − g } . Thus, ( 29 ) con tains the v anishing factor (1 − t Φ 0 ) = (1 − t φ ′ 0 ( σ o , σ ′ o ) ) = (1 − t σ o, 0 ) = 0, which completes the pro of. □ F or a pair of distinct conﬁgurations  σ  =  σ ′ ∈ S ( m ), we say that  σ ′ is admissible to  σ if the conditions in T able 1, or equiv alen tly (i) and (ii), are satisﬁed, and in addition, σ ′ o, 0 ≥ 1 holds. In view of Prop osition 1 , the expansion of H n,l |  σ ⟩ contains a term |  σ ′ ⟩ with nonzero co eﬃcien t if and only if  σ ′ is admissible to  σ . Remark 2. Let us consider the sp ecial case l = 1 of ( 29 ). As for the site o , the relation ( 25 ) holds for µ = 2, and r 1 and r 2 are the ﬁnal and the initial lo cal states at site o , resp ectiv ely . Since σ o,h = 0 for h ∈ [0 , r 1 ) ⊔ ( r 2 , n ], we hav e φ ′ h ( σ o , σ ′ o ) = 0. The relation ( 27 ) holds for µ = 2, and r 1 and r 2 are the initial and the ﬁnal lo cal states at site j (  = o ), resp ectiv ely . Since σ j,h = 0 for h ∈ ( r 1 , r 2 ), we hav e φ h ( σ j , σ ′ j ) = 0 for j  = o . It follows that Φ 0 = · · · = Φ n = 0 for l = 1 . (31) Supp ose that 0 ∈ { h 0 , . . . , h g } . Then, from the assumption 0 ≤ h 0 < · · · < h g ≤ n , it follows that h 0 = 0. Hence, the second pro duct in ( 29 ) b ecomes 1 due to ( 31 ). W e also hav e σ p ( h ) ,h = 1 by ( 19 ), K 0 = 1 from ( 14 ), and ℓ 0 = 0 from ( 22 ). Thus, ( 29 ) reduces to n Y h =0 w ( o )  σ ,  σ ′ ( h ) = Y h ∈{ h 1 ,...,h g } (1 − t ) t ℓ h 1 − t K h , (32) where 1 ≤ h 1 , . . . , h g ≤ n denote the sp ecies of the m o ved particles. The formula ( 32 ) coincides with the kno wn expression given in [ AMW24 , eq. (2.2)] and [ AK25 , eq. (2.7)]. On the other hand, supp ose that 0 / ∈ { h 0 , . . . , h g } , so that 0 ∈ { ¯ h 1 , . . . , ¯ h n − g } . In this case, the second pro duct in ( 29 ) con tains the factor 1 − t Φ 0 1 − t K 0 , which v anishes due to ( 31 ). 4 This result is consistent with the constrain t implied by the dynamics deﬁned in [ AMW24 , AK25 ]. Example 3. F or n = 4 , l = 3 , L = 3, consider the following states in m ultiplicit y and tableau represen tations ( 9 ) in V ( m ) with m = (2 , 3 , 1 , 2 , 1): |  σ ⟩ = | (1 , 1 , 1 , 0 , 0) , (0 , 2 , 0 , 0 , 1) , (1 , 0 , 0 , 2 , 0) ⟩ = | 012 , 114 , 033 ⟩ , |  σ ′ ⟩ = | (0 , 2 , 0 , 1 , 0) , (1 , 1 , 1 , 0 , 0) , (1 , 0 , 0 , 1 , 1) ⟩ = | 113 , 012 , 034 ⟩ , |  σ ′′ ⟩ = | (0 , 2 , 1 , 0 , 0) , (1 , 1 , 0 , 0 , 1) , (1 , 0 , 0 , 2 , 0) ⟩ = | 112 , 014 , 033 ⟩ , |  σ ′′′ ⟩ = | (1 , 2 , 0 , 0 , 0) , (0 , 1 , 1 , 0 , 1) , (1 , 0 , 0 , 2 , 0) ⟩ = | 011 , 124 , 033 ⟩ . They corresp ond to ( K 0 , K 1 , K 2 , K 3 , K 4 ) = (3 , 2 , 5 , 6 , 8). The Marko v matrix acts as H 4 , 3 ( x 1 , x 2 , x 3 ) |  σ ⟩ = t 6 (1 − t ) 3 (1 − t 2 ) x 2 (1 − t 3 )(1 − t 5 )(1 − t 6 )(1 − t 8 ) |  σ ′ ⟩ + t (1 − t )(1 − t 9 ) x 2 (1 − t 3 )(1 − t 8 ) |  σ ′′ ⟩ + t 3 (1 − t ) 2 x 1 (1 − t 3 )(1 − t 5 ) |  σ ′′′ ⟩ · · · , (33) whic h corresp onds to o = 2 , 2 and 1 in ( 17a ), resp ectiv ely . Since σ ′ 2 , 0 = σ ′′ 2 , 0 = σ ′′′ 1 , 0 = 1, the conﬁgurations  σ ′ ,  σ ′′ ,  σ ′′′ are all admissible to  σ in the sense deﬁned after Prop osition 1 . 4 This fact can, of course, also be deduced from Prop osition 1 . 8 AR VIND A YYER AND A TSUO KUNIBA Let us derive the ab o v e expansion from ( 17a ), i.e., H 4 , 3 ( x 1 , x 2 , x 3 ) |  σ ⟩ = 1 x 2 4 Y h =0 w (2)  σ ,  σ ′ ( h ) |  σ ′ ⟩ + 1 x 2 4 Y h =0 w (2)  σ ,  σ ′′ ( h ) |  σ ′′ ⟩ + 1 x 1 4 Y h =0 w (1)  σ ,  σ ′′′ ( h ) |  σ ′′′ ⟩ + · · · . (34) F or the ﬁrst term |  σ ′ ⟩ , we hav e o = 2, g = 4 and ( h 0 , h 1 , h 2 , h 3 , h 4 ) = (0 , 1 , 2 , 3 , 4) , ( p ( h 0 ) , p ( h 1 ) , p ( h 2 ) , p ( h 3 ) , p ( h 4 )) = (1 , 2 , 1 , 3 , 2) , ( ℓ 0 , ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 ) = (0 , 1 , 2 , 1 , 2) . Th us ( 29 ) is calculated as 4 Y h =0 w (2)  σ ,  σ ′ ( h ) = (1 − t σ 1 , 0 ) t ℓ 0 1 − t K 0 (1 − t σ 2 , 1 ) t ℓ 1 1 − t K 1 (1 − t σ 1 , 2 ) t ℓ 2 1 − t K 2 (1 − t σ 3 , 3 ) t ℓ 3 1 − t K 3 (1 − t σ 2 , 4 ) t ℓ 4 1 − t K 4 = 1 − t 1 − t 3 (1 − t 2 ) t 1 − t 2 (1 − t ) t 2 1 − t 5 (1 − t 2 ) t 1 − t 6 (1 − t ) t 2 1 − t 8 = t 6 (1 − t ) 3 (1 − t 2 ) (1 − t 3 )(1 − t 5 )(1 − t 6 )(1 − t 8 ) . (35) F or the second term |  σ ′′ ⟩ , we hav e o = 2, g = 1 and ( h 0 , h 1 ) = (0 , 1) , ( p ( h 0 ) , p ( h 1 )) = (1 , 2) , ( ℓ 0 , ℓ 1 ) = (0 , 1) , (Φ 2 , Φ 3 , Φ 4 ) = (0 , 0 , 1) . In particular, Φ 4 = φ ′ 4 ( σ 2 , σ ′′ 2 ) + φ 4 ( σ 1 , σ ′′ 1 ) + φ 4 ( σ 3 , σ ′′ 3 ) = 1 is ev aluated as φ 4 ( σ 1 , σ ′′ 1 ) = 0 : σ 1 − σ ′′ 1 = e r 1 − e r 2 , ( r 1 , r 2 ) = (0 , 1) , 4 ∈ (0 , 1) in ( 26 ) , φ ′ 4 ( σ 2 , σ ′′ 2 ) = 1 : σ ′′ 2 − σ 2 = e r 1 − e r 2 , ( r 1 , r 2 ) = (0 , 1) , 4 ∈ [0 , 0) ⊔ (1 , 4] , σ 2 , 4 = 1 in ( 24 ) , φ 4 ( σ 3 , σ ′′ 3 ) = 0 : σ 3 = σ ′′ 3 . Th us ( 29 ) is calculated as 4 Y h =0 w (2)  σ ,  σ ′′ ( h ) = (1 − t σ 1 , 0 ) t ℓ 0 1 − t K 0 (1 − t σ 2 , 1 ) t ℓ 1 1 − t K 1 1 − t K 2 +Φ 2 1 − t K 2 1 − t K 3 +Φ 3 1 − t K 3 1 − t K 4 +Φ 4 1 − t K 4 = 1 − t 1 − t 3 (1 − t 2 ) t 1 − t 2 1 − t 5 1 − t 5 1 − t 6 1 − t 6 1 − t 9 1 − t 8 = t (1 − t )(1 − t 9 ) (1 − t 3 )(1 − t 8 ) . (36) F or the third term |  σ ′′′ ⟩ , we hav e o = 1, g = 1 and ( h 0 , h 1 ) = (1 , 2) , ( p ( h 0 ) , p ( h 1 )) = (2 , 1) , ( ℓ 1 , ℓ 2 ) = (1 , 2) , (Φ 0 , Φ 3 , Φ 4 ) = (1 , 0 , 0) . In particular, Φ 0 = φ ′ 0 ( σ 1 , σ ′′′ 1 ) = 1 originates in σ ′′′ 1 − σ 1 = e 1 − e 2 , ( r 1 , r 2 ) = (1 , 2) , 0 ∈ [0 , 1) ⊔ (2 , 4] , σ 1 , 0 = 1 . Th us ( 29 ) is ev aluated as 4 Y h =0 w (1)  σ ,  σ ′′′ ( h ) = (1 − t σ 2 , 1 ) t ℓ 1 1 − t K 1 (1 − t σ 1 , 2 ) t ℓ 2 1 − t K 2 1 − t Φ 0 1 − t K 0 1 − t K 3 +Φ 3 1 − t K 3 1 − t K 4 +Φ 4 1 − t K 4 = (1 − t 2 ) t 1 − t 2 (1 − t ) t 2 1 − t 5 1 − t 1 − t 3 = t 3 (1 − t ) 2 (1 − t 3 )(1 − t 5 ) . (37) 2.3. Com binatorial description of the transition rates. Let  σ b e a conﬁguration of the n -sp ecies capacit y- l t -PushT ASEP with particle multiplicit y given by m = ( m 0 , . . . , m n ). As shown in Figure 1 , we can think of  σ = ( σ 1 , . . . , σ L ) with σ j = ( σ j, 0 , . . . , σ j,n ) as a matrix with n + 1 ro ws indexed b y i = 0 , 1 , . . . , n from b ottom to top and the L columns indexed by j = 1 , 2 , . . . , L from the left to the right, where σ j,i at the p osition ( i, j ) denotes the num b er of particles of sp ecies i at site j . As an example, let n = 4 , l = 3 , L = 3 , m = (2 , 3 , 1 , 2 , 1). Then a conﬁguration  σ (or a corresponding state denoted by |  σ ⟩ ) is expressed in a matrix form as |  σ ⟩ = | (1 , 1 , 1 , 0 , 0) , (0 , 2 , 0 , 0 , 1) , (1 , 0 , 0 , 2 , 0) ⟩ = | 012 , 114 , 033 ⟩ =     0 1 0 0 0 2 1 0 0 1 2 0 1 0 1     . (38) MUL TISPECIES t -PUSHT ASEP 9 Recall from Figure 1 that transitions can be interpreted as paths on  σ starting from the top row and lea ving from the b ottom row, where we imp ose p eriodic b oundary conditions horizontally . It will b e conv enien t for us here to in terpret  σ instead as an inﬁnite p erio dic array of height n + 1, where σ j + L,i = σ j,i for all 0 ≤ i ≤ n, j ∈ N . F or the ab o v e example, w e will write |  σ ⟩ =     0 1 0 0 1 0 0 1 0 . . . 0 0 2 0 0 2 0 0 2 . . . 1 0 0 1 0 0 1 0 0 . . . 1 2 0 1 2 0 1 2 0 . . . 1 0 1 1 0 1 1 0 1 . . .     . W e now consider down-righ t paths P (dra wn in blue) that satisfy the following conditions: • they start and end at the same column mo dulo L ov erlaid on  σ , • they can turn right w ard 5 at ( i, j ) only if σ j,i ≥ 1, • they can turn down w ard at ( i, j ) only if σ j,i ≤ l − 1, • they can mov e horizontally in a single row for at most L − 1 steps. (39) These conditions imply that the set of such paths is ﬁnite. They are in one-to-one corresp ondence with the p ossible transitions  σ →  σ ′ b y the rule explained in Figure 1 . W e will enclose the entries strictly b elo w this path by a red Y oung diagram Y in F rench notation. F or example, 0 1 0 0 1 0 0 1 0 . . . 0 0 2 0 0 2 0 0 2 . . . 1 0 0 1 0 0 1 0 0 . . . 1 2 0 1 2 0 1 2 0 . . . 1 0 1 1 0 1 1 0 1 . . .                                                 (40) In case P ends up v ertically b elo w where it starts, the Y oung diagram is empt y , i.e., Y = ∅ . W e assign weigh ts to paths as follo ws. Lo cally , the path at a given entry σ j,i = a can lo ok like one of the four conﬁgurations sho wn in Figure 2 , and to each conﬁguration, the weigh t of that entry is shown. a wt( a ) = 1 − t a + K i a wt( a ) = 1 a wt( a ) = 1 − t a a wt( a ) = 1 Figure 2. W eights of lo cal conﬁgurations of blue paths at the cell containing a ∈ [0 , l ]. In the leftmost case, the index i ∈ [0 , n ] is the row to which the cell b elongs, and K i is deﬁned in ( 14 ). In the third and the fourth cases, a = 0 and a = l are forbidden, respectively by ( 39 ). 5 “Right ward” means to the righ t in the ﬁgure , not from the viewp oin t of the tra veler along the path. 10 AR VIND A YYER AND A TSUO KUNIBA Notice that each row of the matrix will only contribute one nontrivial weigh t. The weigh t of the path P is then given by wt( P ) = t P Y Y a ∈ P wt( a ) , (41) where P Y means the sum of all entries in the Y oung diagram Y asso ciated to P . F or example, the conﬁgu- ration in ( 40 ) has ( K 0 , K 1 , K 2 , K 3 , K 4 ) = (0 , 2 , 5 , 6 , 8) and its weigh t is t 3 (1 − t 8 )(1 − t 6 )(1 − t )(1 − t 2 )(1 − t ). Supp ose that the transition  σ →  σ ′ corresp onds to a path P in the sense illustrated in Figure 1 . Then the p o w er P h ∈{ h 0 ,...,h g } ℓ h of t in ( 29 ), ( 22 ) is equal to P Y . The ab o v e deﬁnition of wt( P ) provides a restatemen t of the transition rate ( 29 ) as n Y h =0 w ( o )  σ ,  σ ′ ( h ) = wt( P ) D m , (42) where o sp eciﬁes the column from whic h the path P enters the diagram from the top, and D m is deﬁned in ( 15 ). In the sp ecial case  σ ′ =  σ , the path P runs straight down the diagram, as men tioned earlier, yielding the w eight Q n h =0 (1 − t K h + σ o,h ), whic h coincides with the numerator of ( 17b ). Therefore, the formula ( 17a ) can b e expressed more compactly as H n,l |  σ ⟩ = X  σ ′ ∈S ( m ) L X o =1 1 x o Y 0 ≤ h ≤ n w ( o )  σ ,  σ ′ ( h ) |  σ ′ ⟩ −  L X o =1 1 x o  |  σ ⟩ , (43) = L X o =1 1 x o 1 D m X P ( o ) wt( P ) |  σ ′ ⟩ − |  σ ⟩ ! , (44) where P ( o ) P denotes the sum running ov er all paths entering the diagram at column o . Remark 4. Our mo del is well deﬁned at t = 0. Indeed, from ( 41 ), the blue paths are restricted to those satisfying P Y = 0. T aking the rules ( 39 ) into account, the t = 0 dynamics is expressed as H n,l | t =0 |  σ ⟩ = X 1 ≤ j ≤ L σ j, 0 =0 1 x j  |  σ ( j ) ⟩ − |  σ ⟩  . (45) Here  σ ( j ) is the unique state determined from  σ = ( σ 1 , . . . , σ L ) with σ m = ( σ m, 0 , . . . , σ m,n ) ∈ B l ( 6 ), and the site index j satisfying σ j, 0 = 0. It corresp onds to the unique blue path that enters column j v ertically from the top and exits column j + L vertically from the bottom. In eac h column, the path proceeds down ward un til it reaches the low est cell inscrib ed with a p ositiv e integer, where it turns to the right. If all cells b elo w the entry point in that column are inscrib ed with 0, the path simply contin ues horizontally . A t the ﬁnal step, the path enters column j + L from the left at the b ottom cell inscribed with σ j, 0 = 0, and exits from that p osition after turning down w ard. In terpreting σ m,i as the num b er of particles of species i ∈ [0 , n ] at site m , the transition  σ →  σ ( j ) asso ciated with the ab o v e blue path can b e describ ed as follows. A particle of the smallest av ailable sp ecies (necessarily ≥ 1 under the assumption σ j, 0 = 0) is activ ated at site j . It mov es cyclically to the nearest site on its righ t that contains particles of smaller sp ecies, where it activ ates and bumps out a smallest suc h particle, whic h then contin ues the same pro cedure. This sequential pro cess contin ues until the motion returns to site j after one full wrap around the lattice, where the last activ ated particle, necessarily of sp ecies 0, reﬁlls that site. The t = 0 dynamics is not irreducible in general. F or example, for n = l = 2 and L = 3, there is an inv ariant subspace spanned by | 12 , 02 , 11 ⟩ , | 02 , 12 , 11 ⟩ , and | 12 , 12 , 01 ⟩ (in tableau representation) within V ( m ) with m = (1 , 3 , 2) since the 2’s are untouc hed (dim V ( m ) = 15). A general result on irreducibilit y will b e given in Prop osition 28 . 2.4. Probabilit y conserv ation. Here we sho w, by a direct manipulation, that the Mark ov matrix H n,l indeed conserves the total probabilit y . An alternative pro of based on the integrabilit y of the underlying v ertex mo del will b e given around ( 154 ). MUL TISPECIES t -PUSHT ASEP 11 Set wt j (  σ ) = X P ( j ) wt( P ) . (46) In view of ( 44 ), the following result ensures the total probability conserv ation for the Marko v matrix ( 44 ). Theorem 5. L et  σ b e a c onﬁgur ation of the n -sp e cies c ap acity- l t -PushT ASEP with p article multipicity given by m = ( m 0 , . . . , m n ) . Then wt j (  σ ) = D m for al l 1 ≤ j ≤ L . The pro of of this result will follow from a slightly more general lemma, whic h we now state and prov e. Let n, L b e as ab o v e. Let  τ = ( τ j,i ) b e a perio dic array of nonnegative in tegers, where 0 ≤ i ≤ n and j ∈ N and the p eriodicity condition is as b efore, namely τ j + L,i = τ j,i . In addition, deﬁne m i = P L j =1 τ j,i for 0 ≤ i ≤ n − 1, m = ( m 0 , . . . , m n ) and l = P n i =0 τ 1 ,i . (The undeﬁned m n can b e arbitrary since D m app earing in Lemma 6 b elo w do es not dep end on it. See ( 14 )–( 15 ).) Notice that these conditions mean that there is no constraint on the en tries τ 2 ,n , . . . , τ L,n . W e then deﬁne the set of paths ov erlaid on  τ and deﬁne wt j (  τ ) exactly as ab o v e. Lemma 6. L et  τ b e an inte ger matrix describ e d ab ove. Then wt 1 (  τ ) = D m . Pr o of. W e will p erform induction on n . F or n = 0,  τ can b e written as  τ = ( l , τ 2 , 0 , . . . τ L, 0 , l, τ 2 , 0 , . . . τ L, 0 , . . . ). Then there is a single path P starting ab o v e column 1 and that is forced to leav e b elo w column 1 as w ell. By Figure 2 , wt( P ) = 1 − t l = D ( m 0 ) , proving the result in this case. No w supp ose the result holds for all conﬁgurations with rows 0 , . . . , n − 1, and we wan t to prov e the result for a conﬁguration  τ with rows indexed b y 0 , . . . , n . Consider a path P entering ab ov e the entry τ 1 ,n . There are tw o p ossibilities: either it go es straight down, or it turns righ t at the p oint. Supp ose ﬁrst that it go es straigh t down. Then, by the induction hypothesis, the w eight of any such path is 1 − t τ 1 ,n + K n times D m ′ , where m ′ = ( m 0 , . . . , m n − 1 ). T o b e precise, letting l ′ = l − τ 1 ,n , the total weigh t of all such paths is (1 − t τ 1 ,n + m 0 + ··· + m n − 1 )(1 − t l ′ ) n − 2 Y i =0 (1 − t m 0 + ··· + m i ) . (47) Note that we hav e used the fact that the sum ov er all such entries in the ﬁrst column is l ′ . W e observe that if τ 1 ,n = 0, then the path has to necessarily go down at p osition (1 , n ). In that case, ( 47 ) is the total weigh t of all such paths and it is equal to D m since l ′ = l . No w, supp ose the path turns right at p osition (1 , n ), where we are assuming τ 1 ,n ≥ 1. The weigh t of that cell is 1 − t τ 1 ,n . By the fourth prop ert y of the path, it must turn down w ard at p osition ( k , n ) for some k ≤ L . Therefore, the Y oung diagram Y necessarily contains the rectangle bounded by the points (1 , 0) , (1 , n − 1) , ( k − 1 , 0) and ( k − 1 , n − 1). W e now apply the induction hypothesis to all paths starting at p osition ( k , n − 1). Let l ( k ) = τ k, 0 + · · · + τ k,n − 1 . Notice that the sum of all the entries in the rectangle is P k − 1 j =1 l ( j ) . Then the sum ov er all such paths gives t P k − 1 j =1 l ( j ) (1 − t τ 1 ,n )(1 − t l ( k ) ) n − 2 Y i =0 (1 − t m 0 + ··· + m i ) = (1 − t τ 1 ,n ) n − 2 Y i =0 (1 − t m 0 + ··· + m i )  t P k − 1 j =1 l ( j ) − t P k j =1 l ( j )  . (48) W e now sum ( 48 ) o ver k ranging from 2 to L , noticing that the last factor gives a telescoping sum to get (1 − t τ 1 ,n ) n − 2 Y i =0 (1 − t m 0 + ··· + m i )  t l (1) − t P L j =1 l ( j )  = (1 − t τ 1 ,n ) n − 2 Y i =0 (1 − t m 0 + ··· + m i )  t l ′ − t m 0 + ··· + m n − 1  , (49) where the last p ow er of t is the sum ov er all entries of  τ in rows 0 through n − 1, and we hav e used l (1) = l ′ . T o complete the pro of, we add ( 47 ) and ( 49 ) to get n − 2 Y i =0 (1 − t m 0 + ··· + m i )  (1 − t τ 1 ,n + m 0 + ··· + m n − 1 )(1 − t l ′ ) + (1 − t τ 1 ,n )( t l ′ − t m 0 + ··· + m n − 1 )  , and the sum inside the right paren thesis gives exactly (1 − t τ 1 ,n + l ′ )(1 − t m 0 + ··· + m n − 1 ), which is what we need. □ Pr o of of The or em 5 . The conditions in Lemma 6 w ork for all  σ ∈ S ( m ) and th us, the result go es through. □ 12 AR VIND A YYER AND A TSUO KUNIBA 3. R -ma trix S ( z ) on (antisymmetric ⊗ symmetric) tensor represent a tion 3.1. 3D construction of the quan tum R matrix R ( z ) . F or a, b, i, j ∈ { 0 , 1 } and m ∈ Z ≥ 0 , deﬁne L a,b,m ′ i,j,m b y L 0 , 0 ,m ′ 0 , 0 ,m = L 1 , 1 ,m ′ 1 , 1 ,m = δ m ′ m , L 1 , 0 ,m ′ 1 , 0 ,m = ( − q ) m δ m ′ m , L 0 , 1 ,m ′ 0 , 1 ,m = q ( − q ) m δ m ′ m , L 1 , 0 ,m ′ 0 , 1 ,m = δ m ′ m +1 , L 0 , 1 ,m ′ 1 , 0 ,m = (1 − q 2 m ) δ m ′ m − 1 , (50) and the “weigh t conserv ation” prop ert y L a,b,m ′ i,j,m = 0 unless ( a + b, b + m ′ ) = ( i + j, j + m ) . (51) F or 0 ≤ k ≤ n + 1 and l ≥ 1, we introduce the linear map R ( z ) = R k l ( z , q ) ∈ End( V k ⊗ V l ) via its matrix elemen ts in the multiplicit y representation as follows: R ( z )( v i ⊗ v j ) = X a ∈ B k , b ∈ B l R ( z ) a , b i , j v a ⊗ v b ( i ∈ B k , j ∈ B l ) , (52a) R ( z ) a , b i , j = X α 0 ,...,α n =0 , 1 z α 0 L α 0 ,a n ,b n α n ,i n ,j n L α n ,a n − 1 ,b n − 1 α n − 1 ,i n − 1 ,j n − 1 · · · L α 1 ,a 0 ,b 0 α 0 ,i 0 ,j 0 . (52b) The v ectors v i and v j ha ve b een introduced in ( 5 ) and ( 6 ). This is so-called trace reduction of the 3D L - op erator ov er the ﬁrst comp onen t (cf. [ K22 , eq.(11.44)]), whic h is known to yield the quantum R matrix on V k ⊗ V l regarded as the tensor pro duct of the degree- k antisymmetric tensor representation and the degree- l symmetric tensor representation of U q 2 ( b sl n +1 ) [ K22 , Th.11.5]. As we demonstrate shortly , this formulation allo ws one to compute matrix elements far more eﬃciently than via the con ven tional fusion pro cedure. Example 7. Consider the simplest case k = l = 1. Let i, j ∈ { 0 , . . . , n } and assume i  = j . Then the nonzero elemen ts ( 52b ) are given by R ( z ) e i , e i e i , e i = z − q 2 , R ( z ) e i , e j e i , e j = q (1 − z ) , R ( z ) e i , e j e j , e i = z θ ( ij ) (1 − z ) , S ( z ) e i , e j e j , e i = z θ ( i>j ) (1 − t ) . The elemen t S ( z ) e a , e b e i , e j here coincides with S ( z ) ab ij in [ AK25 , Eq. (3.6)] and with (1 − tz ) R ( z ) n − a,n − b n − i,n − j in [ K OS24 , Eq. (16)]. MUL TISPECIES t -PUSHT ASEP 13 F rom ( 52b ) and ( 53b ), S ( z ) = S k l ( z ) is a ﬁrst-order p olynomial in z for general k and l . Thus, its structure is fully determined by sp ecifying the v alues at z = 0 and the deriv ativ e at z = 0: S ( z ) a , b i , j = S (0) a , b i , j + z ˙ S (0) a , b i , j , ˙ S (0) a , b i , j = d dz S ( z ) a , b i , j     z =0 . (56) They are explicitly given in Prop ositions 9 b elo w. In what follows, we use the notation i Ω = X u ∈ Ω i u , ( j a ) Ω = X u ∈ Ω j u a u , for interv als of the form Ω = ( r, s ) , [ r, s ) , ( s, r ] , [ r , s ], and similarly for a Ω , j Ω , ( ij ) Ω , etc. Prop osition 9. L et 0 ≤ k ≤ n and l ≥ 1 , and supp ose that a , i ∈ B k and b , j ∈ B l ar e multiplicity r epr esentations satisfying a + b = i + j . Then, S (0) a , b i , j is given by S (0) a , b i , j = θ ( j ⪯ b )( − 1) α t β (1 − t j r 1 )(1 − t j r 3 ) · · · (1 − t j r µ − 1 ) , (57) α = i ( r 1 ,r 2 ) + i ( r 3 ,r 4 ) + · · · + i ( r µ − 1 ,r µ ) , (58) β = X 0 ≤ r n + 1. The form ulas ( 81 ) and ( 82 ) reduce to [ AK25 , eqs. (4.5), (4.6)] when l = 1. W e write the deriv ative of the transfer matrices simply as ˙ T k ( z ) = dT k ( z ) dz . It is not diagonal in general, but the calculation of the diagonal elements is easy by using ( 80 ). The results read ⟨  σ | ˙ T k (0) |  σ ⟩ = − L X j =1 1 x j e k ( t K 0 + σ j, 0 , t K 1 + σ j, 1 , . . . , t K n + σ j,n ) . (86) Example 10. W e set n = 2 , l = 2 , L = 3, and consider the transfer matrices acting on the sector V ( m ) with m = (3 , 2 , 1). Then, from ( 14 ), we ha ve ( K 0 , K 1 , K 2 ) = (0 , 3 , 5). W e adopt the tableau representation in ( 9 ) for the lo cal states from B 2 as 00 = (2 , 0 , 0), 01 = (1 , 1 , 0), 02 = (1 , 0 , 1), 11 = (0 , 2 , 0) and 12 = (0 , 1 , 1). MUL TISPECIES t -PUSHT ASEP 17 The action of the transfer matrices is given by: T 0 ( z ) | 12 , 00 , 01 ⟩ = ( − z + x 1 )( − z + x 2 )( − z + x 3 ) x 1 x 2 x 3 | 12 , 00 , 01 ⟩ , T 1 ( z ) | 12 , 00 , 01 ⟩ = (1 − t ) 2 t 3 z ( − z + x 2 ) x 1 x 2 | 01 , 00 , 12 ⟩ + (1 − t ) 2 t 2 z ( − z + x 2 ) x 1 x 2 | 02 , 00 , 11 ⟩ + (1 − t ) 2 t 4 z ( − z + x 2 ) x 1 x 2 | 11 , 00 , 02 ⟩ + (1 − t 2 )(1 − t ) z ( − tz + x 3 ) x 1 x 3 | 02 , 01 , 01 ⟩ + (1 − t 2 )(1 − t ) tz ( − tz + x 3 ) x 1 x 3 | 01 , 02 , 01 ⟩ + (1 − t 2 )(1 − t ) 2 t 2 z 2 x 1 x 3 | 11 , 02 , 00 ⟩ + (1 − t 2 )(1 − t ) tz ( − tz + x 1 ) x 1 x 3 | 12 , 01 , 00 ⟩ + D 1 ( z ) | 12 , 00 , 01 ⟩ , T 2 ( z ) | 12 , 00 , 01 ⟩ = − (1 − t ) 2 (1 − t 2 ) t 4 z x 1 | 01 , 01 , 02 ⟩ + (1 − t ) 2 t 7 z ( − z + x 2 ) x 1 x 2 | 01 , 00 , 12 ⟩ + (1 − t ) 2 t 8 z ( − z + x 2 ) x 1 x 2 | 02 , 00 , 11 ⟩ + (1 − t ) 2 t 4 z ( − t 2 z + x 2 ) x 1 x 2 | 11 , 00 , 02 ⟩ + (1 − t 2 )(1 − t ) t 6 z ( − t 2 z + x 1 ) x 1 x 3 | 12 , 01 , 00 ⟩ + (1 − t 2 )(1 − t ) t 4 z ( − t 2 z + x 3 ) x 1 x 3 | 01 , 02 , 01 ⟩ + (1 − t 2 )(1 − t ) t 6 z ( − tz + x 3 ) x 1 x 3 | 02 , 01 , 01 ⟩ + D 2 ( z ) | 12 , 00 , 01 ⟩ , T 3 ( z ) | 12 , 00 , 01 ⟩ = t 8 ( − t 2 z + x 1 )( − t 2 z + x 2 )( − t 2 z + x 3 ) x 1 x 2 x 3 | 12 , 00 , 01 ⟩ . The co eﬃcien t ( 80 ) of the diagonal term generated by T k ( z ) is denoted by D k ( z ). In particular, the expres- sions for D 0 ( z ) and D 3 ( z ) admit compact forms owing to ( 81 ) and ( 82 ), resp ectiv ely . ˙ T 0 (0) | 12 , 00 , 01 ⟩ = −  1 x 1 + 1 x 2 + 1 x 3  | 12 , 00 , 01 ⟩ , ˙ T 1 (0) | 12 , 00 , 01 ⟩ = (1 − t ) 2 t 3 x 1 | 01 , 00 , 12 ⟩ + (1 − t )(1 − t 2 ) t x 1 | 01 , 02 , 01 ⟩ + (1 − t ) 2 t 2 x 1 | 02 , 00 , 11 ⟩ + (1 − t )(1 − t 2 ) x 1 | 02 , 01 , 01 ⟩ + (1 − t ) 2 t 4 x 1 | 11 , 00 , 02 ⟩ + (1 − t )(1 − t 2 ) t x 3 | 12 , 01 , 00 ⟩ + ˙ D 1 (0) | 12 , 00 , 01 ⟩ , ˙ T 2 (0) | 12 , 00 , 01 ⟩ = (1 − t ) 2 t 7 x 1 | 01 , 00 , 12 ⟩ − (1 − t ) 2 (1 − t 2 ) t 4 x 1 | 01 , 01 , 02 ⟩ + (1 − t )(1 − t 2 ) t 4 x 1 | 01 , 02 , 01 ⟩ + (1 − t ) 2 t 8 x 1 | 02 , 00 , 11 ⟩ + (1 − t )(1 − t 2 ) t 6 x 1 | 02 , 01 , 01 ⟩ + (1 − t ) 2 t 4 x 1 | 11 , 00 , 02 ⟩ + (1 − t )(1 − t 2 ) t 6 x 3 | 12 , 01 , 00 ⟩ + ˙ D 2 (0) | 12 , 00 , 01 ⟩ , ˙ T 3 (0) | 12 , 00 , 01 ⟩ = −  1 x 1 + 1 x 2 + 1 x 3  t 8 | 12 , 00 , 01 ⟩ . Note that the co eﬃcien ts of the oﬀ-diagonal terms in ˙ T 2 (0) are neither all p ositiv e nor all negative in the range 0 < t < 1 and ∀ x j > 0. 18 AR VIND A YYER AND A TSUO KUNIBA According to ( 86 ), the functions ˙ D k (0) = d D k ( z ) dz    z =0 (0 ≤ k ≤ 3) are given as ⟨ 12 , 00 , 01 | ˙ T k (0) | 12 , 00 , 01 ⟩ = − 1 x 1 e k ( t σ 1 , 0 , t K 1 + σ 1 , 1 , t K 2 + σ 1 , 2 ) − 1 x 2 e k ( t σ 2 , 0 , t K 1 + σ 2 , 1 , t K 2 + σ 2 , 2 ) − 1 x 3 e k ( t σ 3 , 0 , t K 1 + σ 3 , 1 , t K 2 + σ 3 , 2 ) = − 1 x 1 e k (1 , t K 1 +1 , t K 2 +1 ) − 1 x 2 e k ( t 2 , t K 1 , t K 2 ) − 1 x 3 e k ( t, t K 1 +1 , t K 2 ) = − 1 x 1 e k (1 , t 4 , t 6 ) − 1 x 2 e k ( t 2 , t 3 , t 5 ) − 1 x 3 e k ( t, t 4 , t 5 ) , (87) where the multiplicit y representations σ 1 = (0 , 1 , 1) , σ 2 = (2 , 0 , 0) and σ 3 = (1 , 1 , 0) are used. 5. Marko v ma trix fr om transfer ma trices Based on the transfer matrices in Section 4 , we in tro duce a linear op erator H n,l = H n,l ( x 1 , . . . , x L ) on V ( m ) dep ending on the parameters x 1 , . . . , x L as an alternating sum H n,l = D − 1 m n +1 X k =0 ( − 1) k − 1 ˙ T k (0) −  L X j =1 1 x j  Id , (88) where D m is deﬁned by ( 15 ). The ﬁrst main result in this pap er is the following: Theorem 11. The Markov matrix H n,l of the n -sp e cies c ap acity- l t -PushT ASEP in ( 17a ) - ( 17b ) is identiﬁe d with H n,l in ( 88 ) . Namely, the fol lowing e quality holds in e ach se ctor V ( m ) : H n,l ( x 1 , . . . , x L ) = H n,l ( x 1 , . . . , x L ) . (89) Example 12. Consider the same case n = l = 2, L = 3 as Example 10 . Then the Marko v matrix ( 17a )-( 17b ) giv es H 2 , 2 | 12 , 00 , 01 ⟩ = (1 − t ) 2 t 3 (1 − t 4 ) (1 − t 2 )(1 − t 3 )(1 − t 5 ) x 1 | 01 , 00 , 12 ⟩ + (1 − t ) 2 t 4 (1 − t 3 )(1 − t 5 ) x 1 | 01 , 01 , 02 ⟩ + (1 − t ) t (1 − t 5 ) x 1 | 01 , 02 , 01 ⟩ + (1 − t ) 2 t 2 (1 − t 6 ) (1 − t 2 )(1 − t 3 )(1 − t 5 ) x 1 | 02 , 00 , 11 ⟩ + (1 − t )(1 − t 6 ) (1 − t 3 )(1 − t 5 ) x 1 | 02 , 01 , 01 ⟩ + (1 − t ) t (1 − t 3 ) x 3 | 12 , 01 , 00 ⟩ −  1 x 1 + t (1 − t ) x 3 (1 − t 3 )  | 12 , 00 , 01 ⟩ . (90) On the other hand, from Example 10 , we get ( − ˙ T 0 (0) + ˙ T 1 (0) − ˙ T 2 (0) + ˙ T 3 (0)) | 12 , 00 , 01 ⟩ = (1 − t ) 2 t 3 (1 − t 4 ) x 1 | 01 , 00 , 12 ⟩ + (1 − t ) 2 t 4 (1 − t 2 ) x 1 | 01 , 01 , 02 ⟩ + (1 − t ) t (1 − t 2 )(1 − t 3 ) x 1 | 01 , 02 , 01 ⟩ + (1 − t ) 2 t 2 (1 − t 6 ) x 1 | 02 , 00 , 11 ⟩ + (1 − t )(1 − t 2 )(1 − t 6 ) x 1 | 02 , 01 , 01 ⟩ + (1 − t ) t (1 − t 2 )(1 − t 5 ) x 3 | 12 , 01 , 00 ⟩ +  (1 − t 2 )(1 − t 3 )(1 − t 5 ) x 2 + (1 − t )(1 − t 4 )(1 − t 5 ) x 3  | 12 , 00 , 01 ⟩ . (91) In particular, the co eﬃcien t of the diagonal term in the last line of ( 91 ) has b een obtained by taking the alternating sum of ( 87 ): 3 X k =0 ( − 1) k − 1  − 1 x 1 e k (1 , t 4 , t 6 ) − 1 x 2 e k ( t 2 , t 3 , t 5 ) − 1 x 3 e k ( t, t 4 , t 5 )  b y means of ( 85 ). Using ( 91 ) and D m = (1 − t 2 )(1 − t 3 )(1 − t 5 ), one can verify that H 2 , 3 | 12 , 00 , 01 ⟩ in ( 88 ) repro duces H 2 , 3 | 12 , 00 , 01 ⟩ as given in ( 90 ). MUL TISPECIES t -PUSHT ASEP 19 The rest of this section is devoted to the pro of of Theorem 11 . 5.1. Diagonal elem en ts. W e ﬁrst pro ve ( 89 ) for the diagonal elements, i.e., ⟨  σ | H n,l |  σ ⟩ = ⟨  σ |H n,l |  σ ⟩ . (92) F rom ( 86 ), ( 85 ) and ( 15 ), the RHS is ev aluated as ⟨  σ |H n,l |  σ ⟩ = D − 1 m n +1 X k =0 ( − 1) k − 1 ⟨  σ | ˙ T k (0) |  σ ⟩ − L X j =1 1 x j = D − 1 m n +1 X k =0 ( − 1) k L X j =1 1 x j e k ( t σ j, 0 , t K 1 + σ j, 1 , . . . , t K n + σ j,n ) − L X j =1 1 x j = L X j =1 1 x j n Y h =0 1 − t K h + σ j,h 1 − t K h − L X j =1 1 x j = L X j =1 C  σ ,j ( t ) − 1 x j , where C  σ ,j ( t ) is deﬁned in ( 17b ). This coincides with ⟨  σ | H n,l |  σ ⟩ , which is the co eﬃcient of |  σ ⟩ in the second term of ( 17a ). 5.2. Reduced diagram and its depth. F rom now on, we assume  σ ′  =  σ and concentrate on the oﬀ- diagonal elements ⟨  σ ′ | H n,l |  σ ⟩ and ⟨  σ ′ |H n,l |  σ ⟩ . The former is given, from ( 17a ), as ⟨  σ ′ | H n,l |  σ ⟩ = L X o =1 ⟨  σ ′ | H n,l |  σ ⟩ o , (93a) ⟨  σ ′ | H n,l |  σ ⟩ o = 1 x o Y 0 ≤ h ≤ n w ( o )  σ ,  σ ′ ( h ) , (93b) where the factor w ( o )  σ ,  σ ′ ( h ) has b een deﬁned in ( 21 ). On the other hand ⟨  σ ′ |H n,l |  σ ⟩ is given, from ( 77b ) and ( 88 ), as ⟨  σ ′ |H n,l |  σ ⟩ = D − 1 m n +1 X k =0 ( − 1) k − 1 L X o =1 ⟨  σ ′ | ˙ T k (0) |  σ ⟩ o , (94a) ⟨  σ ′ | ˙ T k (0) |  σ ⟩ o = 1 x o X a 1 ,..., a L ∈ B k S k l (0) a 2 , σ ′ 1 a 1 , σ 1 · · · ˙ S k l (0) a o +1 , σ ′ o a o , σ o · · · S k l (0) a 1 , σ ′ L a L , σ L . (94b) where ˙ S ( z ) = dS ( z ) dz . Thus the equality ⟨  σ ′ | H n,l |  σ ⟩ = ⟨  σ ′ |H n,l |  σ ⟩ for any  σ  =  σ ′ ∈ S ( m ) follows once we sho w D − 1 m n +1 X k =0 ( − 1) k − 1 x o ⟨  σ ′ | ˙ T k (0) |  σ ⟩ o = Y 0 ≤ h ≤ n w ( o )  σ ,  σ ′ ( h ) . (95) F rom ( 15 ), ( 29 ) and ( 94b ), this is equiv alen t to n +1 X k =0 ( − 1) k − 1 X a 1 ,..., a L ∈ B k S k l (0) a 2 , σ ′ 1 a 1 , σ 1 · · · ˙ S k l (0) a o +1 , σ ′ o a o , σ o · · · S k l (0) a 1 , σ ′ L a L , σ L = Y h ∈{ h 0 ,...,h g } (1 − t σ p ( h ) ,h ) t ℓ h Y h ∈{ ¯ h 1 ,..., ¯ h n − g } (1 − t K h +Φ h ) (  σ  =  σ ′ ) . (96) This relation ac hiev es tw o simpliﬁcations from the original problem. There is no summation o v er the sites o = 1 , . . . , L , and the dep endence on x 1 , . . . , x L is eliminated, leaving it dep enden t only on the parameter t . Elemen ts of S k l (0) and ˙ S k l (0) consisting of ( 96 ) hav e b een obtained in Prop osition 9 . Let us depict x o ⟨  σ ′ | ˙ T k (0) |  σ ⟩ o as in Figure 3 , omitting the sp ectral parameters z /x i since they are set to zero. All the v ertical arrows from σ i to σ ′ i with σ i = σ ′ i , corresp onding to lo cal diagonal transitions, are suppressed. Moreov er, for simplicity , we apply a cyclic shift such that the site o app ears in the leftmost p osition, and attach the symbol ◦ to it to indicate that ˙ S (0) is used there, in contrast to S (0) elsewhere. W e 20 AR VIND A YYER AND A TSUO KUNIBA refer to suc h a diagram as r e duc e d diagr am . See ( 97 ), where a i ∈ B k , σ i  = ρ i ∈ B l for 0 ≤ i ≤ g with some 1 ≤ g < L . 6 X a 0 ,..., a g ∈ B k σ 0 ρ 0 a 0 a 1 σ 1 ρ 1 a 2 · · · σ g ρ g a g a 0 (97) The diagram should b e understo od as representing the sum in ( 94b ), where the L − g − 1 v ertical arro ws corresp onding to the diagonal transitions are suppressed, but their asso ciated vertex weigh ts should still b e accoun ted for. Since the carriers a i ’s remain unc hanged when crossing the suppressed vertical arrows, the sum reduces to those ov er a 0 , . . . , a g ∈ B k , where a i +1 = a i + σ i − ρ i ( i mo d g + 1) in multiplicit y represen tation. Lemma 13. ⟨  σ ′ | ˙ T k (0) |  σ ⟩ o = 0 unless the r e duc e d diagr am ( 97 ) for it satisﬁes the c onditions: ρ 0 + · · · + ρ g = σ 0 + · · · + σ g , (98a) ρ 0 ≺ σ 0 , σ 1 ≺ ρ 1 , . . . , σ g ≺ ρ g , (98b) wher e ( 98a ) is an e quality of the arr ays in the multiplicity r epr esentation, and ≺ is deﬁne d in ( 11 ) . Pr o of. F rom weigh t conserv ation, ( 97 ) v anishes unless the condition ( 98a ) holds. The condition ( 98b ) follows rom Prop osition 9 . □ Lemma 14. L et  σ →  σ ′ b e a tr ansition of states in ( B l ) L . Then the fol lowing two c onditions ar e e quivalent: (a) The tr ansition ﬁts the scheme describ e d in T able 1 ; e quivalently, it satisﬁes c onditions (i) and (ii) state d b etwe en ( 20 ) and ( 21 ) . (b) The asso ciate d r e duc e d diagr am ( 97 ) 7 satisﬁes c onditions ( 98a ) and ( 98b ) . Pr o of. The implication (a) ⇒ (b) is straightforw ard by reference to Figure 1 . No w assume (b). Let ρ o − σ o = e r 1 − · · · in accordance with ( 11 ) or ( 25 ), where the ellipsis · · · denotes an alternating sum of some e s with s > r 1 . The same conv en tion applies in the sequel. Then by condition ( 98a ), there must exist a pair σ j ≺ ρ j among those in ( 98b ) such that ρ j = e r 1 − · · · , and all other pairs satisfy σ j ′ − ρ j ′ = e r ′ − · · · with r ′ > r 1 . Rep eating this argument shows that the transition describ ed in ( 97 ) ﬁts the scheme in T able 1 . □ Supp ose the reduced diagram ( 97 ) satisﬁes ( 98a ) and ( 98b ). T o ensure the weigh t c onserv ation at every v ertex, the capacity k of the carriers m ust b e at least a certain v alue. W e deﬁne the minimum p ossible capacit y as the depth d of the reduced diagram or the transition  σ →  σ ′ . Clearly , the depth is unaﬀected b y the diagonal part of the transition which is suppressed in the reduced diagram. W e refer to the carriers whose capacity equals the depth as minimal c arriers . They are uniquely determined from  σ and  σ ′ . See Example 15 b elow. Example 15. Examples of reduced diagrams and minimal carries for n = 4 , l = 2 and depth d = 1 , 2 , 3. W e adopt the tableau representation for elements from B 2 as 00 = (2 , 0 , 0 , 0 , 0), 01 = (1 , 1 , 0 , 0 , 0), 02 = (1 , 0 , 1 , 0 , 0), 03 = (1 , 0 , 0 , 1 , 0), 04 = (1 , 0 , 0 , 0 , 1), 24 = (0 , 0 , 1 , 0 , 1), etc. All the diagrams satisfy the conditions ( 98a ) and ( 98b ) with the unique choice of carriers exhibited also in the (column strict) tableau represen tation. The examples (1u), (2u), (3u) are unwante d , whereas (1w), (2w), (3w) are wante d in the 6 σ 0 , . . . , σ g should b e understo od as a relab eling of the lo cal states σ j 0 , . . . σ j g that undergo nondiagonal transitions with respect to the original site indices. In this con text, the leftmost site 0 in ( 97 ) corresponds to the site o . 7 The indices j of σ j in ( 97 ) do not necessarily matc h those in  σ = ( σ 1 , . . . , σ L ), since the sites undergoing diagonal transitions are suppressed in the reduced diagram. MUL TISPECIES t -PUSHT ASEP 21 sense that the condition ( 30 ) is satisﬁed or not, resp ectiv ely . (1u) d = 1 24 14 1 13 23 2 1 (1w) d = 1 24 02 0 00 04 4 0 (2u) d = 2 24 12 1 3 02 03 3 4 13 24 2 4 1 3 (2w) d = 2 24 02 0 1 02 12 1 4 13 34 0 4 0 1 (3u) d = 3 24 12 1 3 2 3 14 24 2 4 3 24 34 1 4 2 13 14 1 4 2 1 3 (3w) d = 3 24 02 0 3 2 3 00 02 2 4 3 02 03 0 4 2 13 14 0 4 2 0 3 Remark 16. Diagrams suc h as Figure 1 depict transitions of states corresp onding to the reduced dia- gram ( 97 ) associated with minimal carriers. Figure 1 represents the case where a 0 , . . . , a g =4 ∈ B d , with depth d = 4 b eing the n um b er of horizontal lines in tersecting ev ery v ertical slice except at the p ositions j = o, j 1 , . . . , j 4 . The minimal carriers are sp eciﬁed by the heights h q of the horizontal lines in the m ultiplic- it y represen tation ( 5 ) as a 0 = ( θ ( i ∈ { h 0 , h 3 , h 5 , h 7 } )) n i =0 , a 1 = ( θ ( i ∈ { h 3 , h 5 , h 6 , h 9 } )) n i =0 , a 2 = ( θ ( i ∈ { h 2 , h 4 , h 6 , h 8 } )) n i =0 , a 3 = ( θ ( i ∈ { h 2 , h 3 , h 6 , h 8 } )) n i =0 , a 4 = ( θ ( i ∈ { h 1 , h 3 , h 6 , h 7 } )) n i =0 . As Example 15 demonstrates, we hav e d ≤ g in general. 5.3. Con tribution from minim um carrier. F or simplicit y , w e write S k l ( z ) in ( 94b ) as S ( z ) in the rest of this section. Lemma 17. L et  σ →  σ ′ b e a tr ansition whose r e duc e d diagr am ( 97 ) satisﬁes c onditions ( 98a ) and ( 98b ) . L et d denote its depth, and let a 1 , . . . , a L ∈ B d b e the minimum c arrier. 8 L et 0 ≤ h 0 < · · · < h g ≤ n b e the sp e cies of the move d p articles, in ac c or danc e with the scheme of T able 1 , as guar ante e d by L emma 14 . Then the fol lowing e quality holds: S (0) a 2 , σ ′ 1 a 1 , σ 1 · · · ˙ S (0) a o +1 , σ ′ o a o , σ o · · · S (0) a 1 , σ ′ L a L , σ L = ( − 1) d +1 Y h ∈{ h 0 ,...,h g } (1 − t σ p ( h ) ,h ) t ℓ h . (99) Pr o of. F rom Remark 16 , the transition can be depicted b y a diagram as in Figure 1 . The LHS is ev aluated via Prop osition 9 , and it evidently consists of three parts: ( − 1) A t B Q (1 − t # ). W e illustrate the corresp ondence of each part using the example in Figure 1 . The general case pro ceeds analogously . The lattice sites relev an t to the reduced diagram in Figure 1 are 0 , j 1 , . . . , j 4 . • The Q (1 − t # ) part. F rom ( 61 ) with ( 25 ), or from ( 57 ) with ( 27 ), the following contributions arise: ˙ S (0) a o +1 , σ ′ o a o , σ o : (1 − t σ o,r 2 )(1 − t σ o,r 4 ) = (1 − t σ o,h 6 )(1 − t σ o,h 9 ) = (1 − t σ p ( h 6 ) ,h 6 )(1 − t σ p ( h 9 ) ,h 9 ) , S (0) a j 1 +1 , σ ′ j 1 a j 1 , σ j 1 : (1 − t σ j 1 ,s 1 )(1 − t σ j 1 ,s 3 )(1 − t σ j 1 ,s 5 ) = (1 − t σ p ( h 2 ) ,h 2 )(1 − t σ p ( h 4 ) ,h 4 )(1 − t σ p ( h 8 ) ,h 8 ) , S (0) a j 2 +1 , σ ′ j 2 a j 2 , σ j 2 : (1 − t σ j 2 ,t 1 ) = (1 − t σ p ( h 3 ) ,h 3 ) , S (0) a j 3 +1 , σ ′ j 3 a j 3 , σ j 3 : (1 − t σ j 3 ,u 1 )(1 − t σ j 3 ,u 3 ) = (1 − t σ p ( h 1 ) ,h 1 )(1 − t σ p ( h 7 ) ,h 7 ) , S (0) a j 4 +1 , σ ′ j 4 a j 4 , σ j 4 : (1 − t σ j 4 ,v 1 )(1 − t σ j 4 ,v 3 ) = (1 − t σ p ( h 0 ) ,h 0 )(1 − t σ p ( h 5 ) ,h 5 ) . (100) The pro duct of all these factors matc hes the RHS Q h ∈{ h 0 ,...,h 9 } (1 − t σ p ( h ) ,h ). Note that the sites suppressed in the reduced diagram do not contribute to this, since the associated factor is the z = 0 specialization of ( 75 ). 8 The indices of σ i and a i do not necessarily match those in ( 97 ), since the latter is a reduced diagram omitting the sites undergoing diagonal transitions. 22 AR VIND A YYER AND A TSUO KUNIBA • The sign factor ( − 1) A . F rom ( 58 ) and ( 62 ), the following contributions to A arise: ˙ S (0) a o +1 , σ ′ o a o , σ o : 1 + a o, [ r 1 ,r 2 ) + a o, [ r 3 ,r 4 ) , (101) S (0) a j 1 +1 , σ ′ j 1 a j 1 , σ j 1 : a j 1 , ( s 1 ,s 2 ) + a j 1 , ( s 3 ,s 4 ) + a j 1 , ( s 5 ,s 6 ) , S (0) a j 2 +1 , σ ′ j 2 a j 2 , σ j 2 : a j 2 , ( t 1 ,t 2 ) , S (0) a j 3 +1 , σ ′ j 3 a j 3 , σ j 3 : a j 3 , ( u 1 ,u 2 ) + a j 3 , ( u 3 ,u 4 ) , S (0) a j 4 +1 , σ ′ j 4 a j 4 , σ j 4 : a j 4 , ( v 1 ,v 2 ) + a j 4 , ( v 3 ,v 4 ) . Here, for example, a o, [ r 1 ,r 2 ) = P r 1 ≤ κ d . The follo wing argumen t is a natural generalization of Step 2 in [ AK25 , Sec.5.3]. In general, non-minimum carriers are obtained by supplementing common letters that are not c on tained in the minim um carries in tableau represen tation. Recall that in the plot as Figure 1 , the minim um carries are represented as the horizontal line segmen ts whose heigh t signiﬁes the letter from [0 , n ] contained in their tableau representations. (This is explained in Remark 16 .) The supplemented letters can b e depicted as extra horizontal lines that do not ov erlap the existing horizontal line segments. An example of them is shown in Figure 4 . − r 4 + s 6 − s 5 + u 4 − u 3 + r 3 − r 2 + v 4 − v 3 + s 4 − s 3 + t 2 − t 1 + s 2 − s 1 + u 2 − u 1 + v 2 − v 1 + r 1 h 9 h 8 h 7 h 7 h 6 h 5 h 5 h 4 h 3 h 3 h 2 h 1 h 0 h 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · f 1 · · · · · · f 2 · · · · · · f 3 · · · · · · f 4 n 0 o j 1 j 2 j 3 j 4 Figure 4. An example of non-minimum carriers inducing the same transition as T able 1 . Horizon tal black line segments sp ecify the minimum carriers as explained in Remark 16 . The heigh ts f 1 , f 2 , f 3 , f 4 ∈ [0 , n ] \ { h 0 , . . . , h 9 } of the extra red lines correspond to the supplemen ted letters to them. Let  σ →  σ ′ b e the pro cess describ ed in T able 1 , where particles of sp ecies h 0 , . . . , h g are mo v ed. Let a ′ 1 , . . . , a ′ L ∈ B k with k > d b e the non-minimum carriers obtained by supplementing distinct num b ers f 1 , . . . , f k − d ∈ { ¯ h 1 , . . . , ¯ h n − g } = [0 , n ] \ { h 0 , . . . , h g } to the minimum carriers a 1 , . . . , a L ∈ B d . See ( 20 ). Set a j = ( a j, 0 , . . . , a j,n ). This construction is stated as a j,f 1 = · · · = a j,f k − d = 0 , a ′ j = a j + e f 1 + · · · + e f k − d (105) for all sites j ∈ { 1 , . . . , L } , where e j is deﬁned in ( 7 ). 24 AR VIND A YYER AND A TSUO KUNIBA Lemma 18. Under the deﬁnition ( 105 ) , the pr o duct in the LHS of ( 99 ) for the non-minimum c arries a ′ 1 , . . . , a ′ L ∈ B k is given by S (0) a ′ 2 , σ ′ 1 a ′ 1 , σ 1 · · · ˙ S (0) a ′ o +1 , σ ′ o a ′ o , σ o · · · S (0) a ′ 1 , σ ′ L a ′ L , σ L = ( − 1) d +1 Y h ∈{ h 0 ,...,h g } (1 − t σ p ( h ) ,h ) t ℓ h Y f ∈{ f 1 ,...,f k − d } t K f +Φ f , (106) wher e Φ f = Φ f (  σ ,  σ ′ ) is deﬁne d in ( 23 ) – ( 27 ) . Pr o of. W e in vestigate the eﬀect of introducing the red lines as in Figure 4 on the three types of factors considered individually in the pro of of Lemma 17 . The relev ant formulas are ( 57 )–( 64 ). • The factor of the form Q (1 − t # ). There is no change in the diﬀerence  σ ′ −  σ , hence such factors in ( 57 ) and ( 61 ) neither change. • The sign factor. Let us illustrate the eﬀect of red lines on ( 58 ) and ( 62 ) along the example in Figure 4 . F rom ( 62 ), w e hav e an extra sign for each red line passing through site o in the vertical segment of heights in [ r 1 , r 2 ) and [ r 3 , r 4 ). F rom ( 58 ), we also hav e an extra sign for each crossing of a red line and a vertical black line. Obviously , their num b ers are even and the pro duct is +1. F rom ( 75 ), diagonal lo cal transitions do not con tribute a sign factor even in the presence of red lines. • The pow er of t . The relev ant quantities are ( 59 ) and ( 63 ). In both of them, the ﬁrst term P 0 j ) (1 − z ) 1 − tz , S ( z ) e i , e j e j , e i = z θ ( i>j ) (1 − t ) 1 − tz (0 ≤ i  = j ≤ n ) . (115) They satisfy X 0 ≤ a,b ≤ n S ( z ) e a , e b e i , e j = 1 (0 ≤ i, j ≤ n ) , (116) S (1) e a , e b e i , e j = δ a j δ b i , (117) where the ﬁrst one is called sum-to-unity condition. The most fundamental prop ert y of the CTM op erators A 0 ( z ) , . . . , A n ( z ) is that they satisfy the Zamolo d- c hiko v–F addeev (ZF) algebra, whose structure function is given by the basic sto c hastic R -matrix ( 115 ): Prop osition 20. [ KOS24 , Th. 28] A b ( x ) A a ( y ) = X 0 ≤ i,j ≤ n S  y x  e a , e b e i , e j A i ( y ) A j ( x ) (0 ≤ a, b ≤ n ) . (118) 6.4. ZF algebra with structure function S k,l ( z ) . Let us generalize the result of the previous subsection to the case where the structure function is replaced b y S k,l ( z ), obtained through the symmetric fusion of the fundamen tal one S 1 , 1 ( z ), as describ ed in App endix A . W e b egin by in tro ducing the op erator A i ( z ) lab eled b y a semistandard tableau i = ( i 1 , . . . , i l ) ∈ T l , of row shap e and length l , whose en tries range in { 0 , . . . , n } ; see ( 9 ). MUL TISPECIES t -PUSHT ASEP 27 Giv en a tableau i = ( i 1 , . . . , i l ) ∈ T l , let C ( i ) denote the set of its distinct p erm utations. F or example, when l = 3 one has C  (0 , 1 , 4)  = { (0 , 1 , 4) , (0 , 4 , 1) , (1 , 0 , 4) , (1 , 4 , 0) , (4 , 0 , 1) , (4 , 1 , 0) } , C  (2 , 3 , 3)  = { (2 , 3 , 3) , (3 , 2 , 3) , (3 , 3 , 2) } , C  (5 , 5 , 5)  = { (5 , 5 , 5) } . (119) The quantized CTM A i ( z ) asso ciated with i ∈ T l is deﬁned by a “fusion” of the basic CTMs as A i ( z ) = X i ′ ∈C ( i ) A i ′ 1 ( t l − 1 z ) A i ′ 2 ( t l − 2 z ) · · · A i ′ l ( z ) ( i ∈ T l ) , (120) where the sum is taken ov er i ′ = ( i ′ 1 , . . . , i ′ l ) ∈ C ( i ). F ormally , this is a t -deformed monomial symmetric p olynomial in A 0 ( z ) , . . . , A n ( z ). It acts linearly on the same space F ⊗ n ( n − 1) / 2 as in the l = 1 case. Example 21. Consider ﬁrst the case n = 1. F rom ( 112 ) we obtain the scalar op erators A 00 ( z ) = A 0 ( tz ) A 0 ( z ) = 1 , A 01 ( z ) = A 0 ( tz ) A 1 ( z ) + A 1 ( tz ) A 0 ( z ) = (1 + t ) z , A 11 ( z ) = A 1 ( tz ) A 1 ( z ) = tz 2 , A 000 ( z ) = 1 , A 001 ( z ) = (1 + t + t 2 ) z , A 011 ( z ) = t (1 + t + t 2 ) z 2 , A 111 ( z ) = t 3 z 3 . In general one easily veriﬁes that A s ( z ) = z r t r ( r − 1) / 2  l r  t for s = (0 , . . . , 0 | {z } l − r , 1 , . . . , 1 | {z } r ) ∈ T l , (121) where  l r  t = ( t ) l ( t ) r ( t ) l − r with ( t ) r = Q r j =1 (1 − t j ) is the t -binomial co eﬃcien t. Example 22. Next consider the case n = 2 and l = 2. Applying ( 120 ) to Example 19 , we obtain A 22 ( z ) = t 2 z 4 + t (1 + t ) z 3 a + + tz 2 a + 2 , A 12 ( z ) = t (1 + t ) z 3 k + t (1 + t ) z 2 a + k , A 11 ( z ) = tz 2 k 2 , A 02 ( z ) = (1 + t ) 2 z 2 + t (1 + t ) z 3 a − + (1 + t ) z a + − t (1 + t ) z 2 k , A 01 ( z ) = (1 + t ) z k + (1 + t ) z 2 a − k , A 00 ( z ) = 1 + (1 + t ) z a − + tz 2 a − 2 . In relation to Remark 34 giv en later, w e note that A 02 ( z ) is also expressible without a min us sign as A 02 ( z ) = (1 + t ) z 2 + t (1 + t ) z 2 a + a − + t (1 + t ) z 3 a − + (1 + t ) z a + b y using ( 111 ). Remark 23. Adding relation ( 118 ) to its counterpart with a and b in terchanged and using ( 116 ), we obtain A b ( x ) A a ( y ) + A a ( x ) A b ( y ) = A b ( y ) A a ( x ) + A a ( y ) A b ( x ) (122) for any 0 ≤ a, b ≤ n . This is regarded as inv ariance under the exchange of x and y . By rep eatedly applying it to neighboring sp ectral parameters, we ﬁnd that for any i ∈ T l , X i ′ ∈C ( i ) A i ′ 1 ( z 1 ) A i ′ 2 ( z 2 ) · · · A i ′ l ( z l ) is inv ariant under an y p erm utation of z 1 , . . . , z l . (123) Th us the sp ectral parameters in ( 120 ) may actually be reordered arbitrarily . The ordering c hosen there is the one suitable for the pro of of the following Theorem 24 . Theorem 24. The op er ators A i ( z ) deﬁne d in ( 120 ) satisfy the Zamolo dchikov-F adde ev algebr a A b ( x ) A a ( y ) = X i ∈ T k , j ∈ T l S k,l  y x  a , b i , j A i ( y ) A j ( x ) ( a ∈ T k , b ∈ T l ) (124) for any k , l ∈ Z ≥ 1 , wher e the structur e function S k,l ( z ) a , b i , j is given in App endix A . 28 AR VIND A YYER AND A TSUO KUNIBA Pr o of. W e illustrate the pro of for the case ( k, l ) = (3 , 2), for which the structure function is depicted in ( 172 ). The general case is completely analogous. F or brevity , we denote S ( z ) e a , e b e i , e j = S 1 , 1 ( z ) e a , e b e i , e j , which app ears in ( 115 ), ( 118 ) and ( 170 ), simply by S ( z ) a,b i,j . Substituting A b ( x ) = X b ′ ∈C ( b ) A b ′ 1 ( tx ) A b ′ 2 ( x ) , A a ( x ) = X a ′ ∈C ( a ) A a ′ 1 ( t 2 y ) A a ′ 2 ( ty ) A a ′ 3 ( y ) in to the LHS of ( 124 ), w e obtain X a ′ , b ′ A b ′ 1 ( tx ) A b ′ 2 ( x ) A a ′ 1 ( t 2 y ) A a ′ 2 ( ty ) A a ′ 3 ( y ) . (125) W e ﬁrst mov e A b ′ 2 ( x ) to the right through A a ′ 1 ( t 2 y ) A a ′ 2 ( ty ) A a ′ 3 ( y ) by successive use of ( 118 ): A b ′ 2 ( x ) A a ′ 1 ( t 2 y ) A a ′ 2 ( ty ) A a ′ 3 ( y ) = X i ′ 1 ,i ′ 2 ,i ′ 3 ,j ′ 2 ,j ′′ 2 ,j ′′′ 2 S ( z t 2 ) a ′ 1 ,b ′ 2 i ′ 1 ,j ′ 2 S ( z t ) a ′ 2 ,j ′ 2 i ′ 2 ,j ′′ 2 S ( z ) a ′ 3 ,j ′′ 2 i ′ 3 ,j ′′′ 2 A i ′ 1 ( t 2 y ) A i ′ 2 ( ty ) A i ′ 3 ( y ) A j ′′′ 2 ( x ) , (126) where z = y /x , and all summation indices range ov er { 0 , . . . , n } . Next, we mov e A b ′ 1 ( tx ) through the three op erators app earing in ( 126 ): A b ′ 1 ( tx ) A i ′ 1 ( t 2 y ) A i ′ 2 ( ty ) A i ′ 3 ( y ) = X i ′′ 1 ,i ′′ 2 ,i ′′ 3 ,j ′ 1 ,j ′′ 1 ,j ′′′ 1 S ( z t ) i ′ 1 ,b ′ 1 i ′′ 1 ,j ′ 1 S ( z ) i ′ 2 ,j ′ 1 i ′′ 2 ,j ′′ 1 S ( z /t ) i ′ 3 ,j ′′ 1 i ′′ 3 ,j ′′′ 1 A i ′′ 1 ( t 2 y ) A i ′′ 2 ( ty ) A i ′′ 3 ( y ) A j ′′′ 1 ( tx ) . (127) Com bining ( 126 ) and ( 127 ), the expression ( 125 ) b ecomes X i ′′ 1 ,i ′′ 2 ,i ′′ 3 A i ′′ 1 ( t 2 y ) A i ′′ 2 ( ty ) A i ′′ 3 ( y ) X j ′′′ 1 ,j ′′′ 2 A j ′′′ 1 ( tx ) A j ′′′ 2 ( x ) W ( z ) a , b ( i ′′ 1 ,i ′′ 2 ,i ′′ 3 ) , ( j ′′′ 1 ,j ′′′ 2 ) , (128) where the co eﬃcient W ( z ) a , b ( i ′′ 1 ,i ′′ 2 ,i ′′ 3 ) ,, ( j ′′′ 1 ,j ′′′ 2 ) admits the following diagrammatic representation (cf. ( 170 )): W ( z ) a , b ( i ′′ 1 ,i ′′ 2 ,i ′′ 3 ) , ( j ′′′ 1 ,j ′′′ 2 ) = X ( a ′ 1 ,a ′ 2 ,a ′ 3 ) ∈ C ( a ) ( b ′ 1 ,b ′ 2 ) ∈ C ( b ) X i ′ 1 ,i ′ 2 ,i ′ 3 j ′ 1 ,j ′′ 1 ,j ′ 2 ,j ′′ 2 i ′ 1 i ′ 2 i ′ 3 j ′ 1 j ′′ 1 j ′ 2 j ′′ 2 z /t z z t z z t z t 2 a ′ 1 a ′ 2 a ′ 3 b ′ 1 b ′ 2 i ′′ 1 i ′′ 2 i ′′ 3 j ′′′ 1 j ′′′ 2 (129) The righ t and left columns corresp ond to ( 126 ) and ( 127 ), resp ectiv ely . F rom Remark 42 , the quan- tit y W ( z ) a , b ( i ′′ 1 ,i ′′ 2 ,i ′′ 3 ) , ( j ′′′ 1 ,j ′′′ 2 ) is in v ariant under p erm utations of ( i ′′ 1 , i ′′ 2 , i ′′ 3 ) and lik ewise under permutations of ( j ′′′ 1 , j ′′′ 2 ). Accordingly , the sum ov er i ′′ 1 , i ′′ 2 , i ′′ 3 can b e reorganized as X i ′′ 1 ,i ′′ 2 ,i ′′ 3 ∈{ 0 ,...,n } = X 0 ≤ i 1 ≤ i 2 ≤ i 3 ≤ n X ( i ′′ 1 ,i ′′ 2 ,i ′′ 3 ) ∈C  ( i 1 ,i 2 ,i 3 )  = X i ∈ T 3 X i ′′ ∈C ( i ) . See ( 9 ) and ( 119 ) for the deﬁnitions of T 3 and C ( i ). T reating the sum ov er j ′′′ 1 , j ′′′ 2 similarly , ( 128 ) can b e written as X i ∈ T 3 X i ′′ ∈C ( i ) A i ′′ 1 ( t 2 y ) A i ′′ 2 ( ty ) A i ′′ 3 ( y ) X j ∈ T 2 X j ′′′ ∈C ( j ) A j ′′′ 1 ( tx ) A j ′′′ 2 ( x ) W ( z ) a , b i ′′ , j ′′′ . (130) MUL TISPECIES t -PUSHT ASEP 29 Comparing the diagrams in ( 129 ) and ( 172 ), and taking Remark 42 into account, w e obtain W ( z ) a , b i ′′ , j ′′′ = S 3 , 2 ( z ) a , b i , j for any i ′′ ∈ C ( i ) and j ′′′ ∈ C ( j ). Thus ( 130 ) equals X i ∈ T 3 , j ∈ T 2 S 3 , 2 ( z ) a , b i , j  X i ′′ ∈C ( i ) A i ′′ 1 ( t 2 y ) A i ′′ 2 ( ty ) A i ′′ 3 ( y )  X j ′′′ ∈C ( j ) A j ′′′ 1 ( tx ) A j ′′′ 2 ( x )  = X i ∈ T 3 , j ∈ T 2 S 3 , 2 ( z ) a , b i , j A i ( y ) A j ( x ) . (131) □ Remark 25. By applying the sum-to-unity prop ert y ( 173 ) of S k,l ( z ) to the ZF-algebra relation ( 124 ), Remark 23 is generalized to arbitrary k and l as follows: X a ∈ T k , b ∈ T l A a ( x ) A b ( y ) = X a ∈ T k , b ∈ T l A a ( y ) A b ( x ) . F rom ( 176 ) and T 1 = T 1 (see ( 8 ), ( 9 )), the sp ecial case k = 1 of Theorem 24 is stated in terms of S 1 l as X i ∈ T 1 , j ∈ T l S 1 l  x w  a , b i , j A i ( t l − 1 x ) A j ( w ) =  1 − t l x w  A b ( w ) A a ( t l − 1 x ) ( a ∈ T 1 , b ∈ T l ) . (132) The following lemma will b e a key ingredient in the pro of of Prop osition 29 . Lemma 26. F or a , i ∈ T 1 and b , j ∈ T l satisfying the weight c onservation (i.e., a + b = i + j in the multiplicity r epr esentation), the fol lowing e quality holds: S 1 l (1) a , b i , j A j ( x ) = ( (1 − t l ) A b \ i ( x ) A a ( t l − 1 x ) if i ⊆ b , 0 otherwise . (133) Her e, i ⊆ b me ans the c ondition b − i ∈ B l − 1 in the multiplicity r epr esentation, and b \ i denotes the element in T l − 1 c orr esp onding to b − i ∈ B l − 1 . Pr o of. Let b = ( b 1 , . . . , b l ) ∈ T l and j = ( j 1 , . . . , j l ) ∈ T l b e tableau represen tations. Below we will freely iden tify the tableau and m ultiplicit y represen tations as explained around ( 10 ). F rom ( 176 ) and the expression for S 1 ,l (1) analogous to ( 172 ), we hav e S 1 l (1) a , b i , j = (1 − t l ) S 1 ,l ( t l − 1 ) a , b i , j = (1 − t l ) X ( b ′ 1 ,...,b ′ l ) ∈C ( b ) X α 1 ,...,α l − 1 ∈ T 1 S (1) α 1 ,b ′ 1 i ,j 1 S ( t ) α 2 ,b ′ 2 α 1 ,j 2 · · · S ( t l − 1 ) a ,b ′ l α l − 1 ,j l , (134) where S ( z ) a , b i , j = S 1 , 1 ( z ) a , b i , j . The set C ( b ) has b een deﬁned in ( 119 ). The op erator A j ( x ) with j ∈ T l is giv en, according to ( 120 ), as A j ( x ) = X ( j ′ 1 ,...,j ′ l ) ∈C ( j ) A j ′ 1 ( t l − 1 x ) · · · A j ′ l ( x ) . (135) F rom Remark 42 , the expression ( 134 ) is inv ariant under replacing ( j 1 , . . . , j l ) ∈ T l b y any array in C ( j ). Cho osing it so as to coincide with ( j ′ 1 , . . . , j ′ l ) in ( 135 ), the LHS of ( 133 ), which is the pro duct of ( 134 ) and ( 135 ), is expressed as (1 − t l ) X ( b ′ 1 ,...,b ′ l ) ∈C ( b ) X α 1 ,...,α l − 1 ∈ T 1 ( j ′ 1 ,...,j ′ l ) ∈C ( j ) S (1) α 1 ,b ′ 1 i ,j ′ 1 S ( t ) α 2 ,b ′ 2 α 1 ,j ′ 2 · · · S ( t l − 1 ) a ,b ′ l α l − 1 ,j ′ l A j ′ 1 ( t l − 1 x ) · · · A j ′ l ( x ) . (136) F rom ( 117 ), the sum is restricted to b ′ 1 = i and α 1 = j ′ 1 . The former condition implies that ( 136 ) v anishes unless i ⊆ b in agreement with ( 133 ). Assuming i ⊆ b in what follows, ( 136 ) is reduced to (1 − t l ) X ( i ,b ′ 2 ,...,b ′ l ) ∈C ( b ) X α 2 ,...,α l − 1 ∈ T 1 ( j ′ 1 ,...,j ′ l ) ∈C ( j ) S ( t ) α 2 ,b ′ 2 j ′ 1 ,j ′ 2 S ( t 2 ) α 3 ,b ′ 3 α 2 ,j ′ 3 · · · S ( t l − 1 ) a ,b ′ l α l − 1 ,j ′ l A j ′ 1 ( t l − 1 x ) A j ′ 2 ( t l − 2 x ) · · · A j ′ l ( x ) . The condition ( i , b ′ 2 , . . . , b ′ l ) ∈ C ( b ) is equiv alen t to ( b ′ 2 , . . . , b ′ l ) ∈ C ( b \ i ). In the pro duct of S app earing here, the total incoming (subscript) w eight j + α 2 + · · · + α l − 1 and the total outgoing (sup erscript) weigh t 30 AR VIND A YYER AND A TSUO KUNIBA b − i + a + α 2 + · · · + α l − 1 coincide by the assumption of the lemma. Under this circumstance, the summation ( j ′ 1 , . . . , j ′ l ) ∈ C ( j ) may b e replaced by the indep endent sums j ′ 1 , . . . , j ′ l ∈ T 1 , since the total w eigh t of ( j ′ 1 , . . . , j ′ l ) is automatically constrained to b e j by the w eigh t conserv ation prop ert y of the individual S ’s. Then, one can apply the ZF-algebra relation ( 118 ) successiv ely to tak e the sums o ver ( j ′ 1 , j ′ 2 ), ( α 2 , j ′ 3 ), ( α 3 , j ′ 4 ), . . . , ( α l − 1 , j ′ l ) in this order, thereby mo ving A • ( t l − 1 x ) through to the righ t, where it even tually becomes A a ( t l − 1 x ) as follows: (1 − t l )   X ( b ′ 2 ,...,b ′ l ) ∈C ( b \ i ) A b ′ 2 ( t l − 2 x ) A b ′ 3 ( t l − 3 x ) · · · A b ′ l ( x )   A a ( t l − 1 x ) . By the deﬁnition ( 120 ), the sum in the parenthesis here is equal to A b \ i ( x ). □ Example 27. F or n = 2 and l = 3, we hav e S 1 3 (1) 2 , 112 1 , 122 = t (1 − t 2 ) , A 122 ( x ) = t 3 (1 + t + t 2 ) x 3 ( tx 2 k + x a + k + tx a + k + a + 2 k ) , A 12 ( x ) = t (1 + t ) x 2 ( x k + a + k ) , A 2 ( x ) = x 2 + x a + . Using ( 111 ), it is straightforw ard to verify that S 1 3 (1) 2 , 112 1 , 122 A 122 ( x ) = (1 − t 3 ) A 12 ( x ) A 2 ( t 2 x ). 7. St a tionar y distribution Here we deriv e our second and third main results, namely the matrix pro duct formula ( 137b ) for the (unnormalized) stationary probabilit y , and the partition function in Theorem 35 . W e begin with the follo wing elemen tary result. Prop osition 28. Supp ose t, x 1 , . . . , x L > 0 . F or any n, l ∈ Z ≥ 1 and m = ( m 0 , . . . , m n ) ∈ ( Z ≥ 1 ) n +1 such that m 0 + · · · + m n = lL , the n -sp e cies c ap acity- l inhomo gene ous t -PushT ASEP on L -site p erio dic chain in the se ctor V ( m ) is irr e ducible. Pr o of. F rom ( 17a ) and ( 29 ), the oﬀ-diagonal elemen ts of the Marko v matrix H n,l = H n,l ( x 1 , . . . , x L ) are p ositiv e under the assumption t, x 1 , . . . , x L > 0. Thus, it suﬃces to show that for any pair of diﬀerent conﬁgurations  σ ,  σ ′ ∈ S ( m ) there is m ∈ Z ≥ 1 suc h that ⟨  σ ′ | ( H n,l ) m |  σ ⟩ > 0. Notice that an y particle of sp ecies greater than 0 can mov e from any site, bumping low er sp ecies particles. T o get  σ ′ from  σ , we ﬁrst mov e the particles of sp ecies n to their desired lo cations by a sequence of mov es. W e then con tin ue this wa y mo ving particles of sp ecies n − 1, and so on. One can also sho w the claim by induction on L . □ F rom Proposition 28 , it follo ws that the stationary distribution is unique. W e no w explain the matrix pro duct formula. 7.1. Matrix product formula. Theorem 11 with ( 88 ) transforms the problem of ﬁnding the stationary state of the n -sp ecies, capacity- l t -PushT ASEP on L -site perio dic lattice with inhomogeneity x 1 , . . . , x L to the study of eigen v ectors of the transfer matrices T 0 ( z | x 1 , . . . , x L ) , . . . , T n +1 ( z | x 1 , . . . , x L ). Thanks to the comm utativity ( 79 ), it is further reduced to ﬁnding the stationary eigenv ector of the transfer matrix T 1 ( z | x 1 , . . . , x L ). This task was done in our previous work [ AK25 ] for the case l = 1 b y a matrix pro duct construction based on the basic CTMs A 0 ( z ) , . . . , A n ( z ). In this section, w e generalize it to arbitrary capacit y l ≥ 1. W e will also provide a deriv ation of the probability conserv ation prop ert y of the Marko v matrix H n,l ( x 1 , . . . , x L ) based on the transfer matrices, which is indep enden t of the argument given in Section 2.4 . Let us introduce a state vector whose co eﬃcien ts are given in the matrix pro duct form: | P ( m ) ⟩ = X ( σ 1 ,..., σ L ) ∈S ( m ) P ( σ 1 , . . . , σ L ) | σ 1 , . . . , σ L ⟩ ∈ V ( m ) , (137a) P ( σ 1 , . . . , σ L ) = T r ( A σ 1 ( x 1 ) · · · A σ L ( x L )) , (137b) where V ( m ) and S ( m ) are deﬁned in ( 12 ) and ( 13 ), resp ectiv ely . The trace is tak en ov er the F o c k space F ⊗ n ( n − 1) / 2 , and is nonzero and con v ergen t under the assumption m 0 , . . . , m n ≥ 1 as a formal p o w er series in t . 11 By the construction, P ( σ 1 , . . . , σ L ) is a p olynomial in x 1 , . . . , x L and a rational function of t . 11 See Remark 34 b elo w for the non v anishing property , and ( 160 ) together with the discussion following [ K OS24 , eq. (59)] for conv ergence in the case l = 1. MUL TISPECIES t -PUSHT ASEP 31 Set Λ k ( z | x 1 , . . . , x L ) = e k − 1 ( t K 1 , . . . , t K n ) L Y j =1  1 − t l z x j  + e k ( t K 1 , . . . , t K n ) L Y j =1  1 − z x j  (0 ≤ k ≤ n + 1) , (138) where e k denotes the elementary symmetric p olynomial ( 84 ), and K i , deﬁned in ( 14 ), dep ends on m . This is a Y ang-Baxterization (sp ectral parameter dep endent version) of the k ’th elementary symmetric p olynomial in the following sense: Λ k ( z | x 1 , . . . , x L ) = L Y j =1 k − 1 Y s =1  1 − t − s z x j  − 1 X 0 ≤ i 1 < ··· 0 and 0 < t < 1. In fact, thanks to ( 160 ), this claim reduces to the case l = 1. It then follows readily from the fact that the CTMs A 0 ( x ) , . . . , A n ( x ) in ( 113 ) are linear com binations of pro ducts of t -oscillator generators with p ositiv e coeﬃcients, and that the matrix elements of the t -oscillators in ( 110 ) are p ositiv e. 7.2. P artition function. W e now fo cus on Z l, m ( x 1 , . . . , x L ; t ) = ⟨ Ω( m ) | P ( m ) ⟩ = X ( σ 1 ,..., σ L ) ∈S ( m ) P l  x 1 , . . . , x L σ 1 , . . . , σ L ; t  , (161) where Ll = | m | = m 0 + · · · + m n . This quan tity is the normalization factor of the stationary probability , often referred to as the p artition function . Here we hav e made the dep endence on t and the capacity l explicit. Due to Remark 34 , the partition function Z l, m ( x 1 , . . . , x L ; t ) is also p ositiv e for x 1 , . . . , x L > 0 and 0 < t < 1. F rom ( 158 ) and the exchange symmetry in Remark 25 , it is a symmetric p olynomial in x 1 , . . . , x L with rational co eﬃcients. Our third main result in this pap er is the following corollary of ( 160 ): Theorem 35. The p artition function with gener al c ap acity l is r e duc e d to the l = 1 c ase as fol lows: Z l, m ( x 1 , x 2 , . . . , x L ; t ) = Z 1 , m  1 − t l 1 − t x 1 , 1 − t l 1 − t x 2 , . . . , 1 − t l 1 − t x L ; t  , (162) wher e (1 − t l ) x j / (1 − t ) c onventional ly denotes the l -tuple of ge ometric al ly weighte d inhomo geneity p ar ameters x j , tx j , . . . , . . . , t l − 1 x j , and the RHS c orr esp onds to a system on a lattic e of length lL with c ap acity 1 . (Se e also the explanation after ( 4 ) .) Example 36. F or n = 2, w e ha ve Z 2 , (1 , 2 , 1) ( x 1 , x 2 ; t ) = t (1 + t ) x 1 x 2 ( x 1 + x 2 ) 2 1 − t , Z 1 , (1 , 2 , 1) ( x 1 , x 2 , x 3 , x 4 ; t ) = e 1 ( x 1 , x 2 , x 3 , x 4 ) e 3 ( x 1 , x 2 , x 3 , x 4 ) 1 − t 2 , where e k denotes the elementary symmetric p olynomial deﬁned in ( 84 ). T hey satisfy the relation ( 162 ), namely , Z 2 , (1 , 2 , 1) ( x 1 , x 2 ; t ) = Z 1 , (1 , 2 , 1) ( x 1 , tx 1 , x 2 , tx 2 ) . In [ CMW22 , AMW24 ], the partition function Z l, m ( x 1 , . . . , x L ; t ) of the t -PushT ASEP with capacity l = 1 and m = ( m 0 , . . . , m n ) has b een link ed to Macdonald polynomials P λ ( x 1 , . . . , x L ; q , t ) at q = 1, where λ = ⟨ n m n , . . . , 1 m 1 , 0 m 0 ⟩ in frequency notation. Combined with the standard factorized formula for the Macdonald p olynomials at q = 1 [ Mac95 , p. 324, (iv)], this yields Z 1 , m ( x 1 , . . . , x L ; t ) = ∆ m ( t ) − 1 n Y i =1 e m i + ··· + m n ( x 1 , . . . , x L ) , (163a) ∆ m ( t ) = Y 1 ≤ i ≤ j ≤ n − 1 (1 − t m i + m i +1 + ··· + m j ) . (163b) 36 AR VIND A YYER AND A TSUO KUNIBA The function ∆ m ( t ), for example 1 , 1 − t m 1 , (1 − t m 1 )(1 − t m 1 + m 2 )(1 − t m 2 ) for n = 1 , 2 , 3, respectively , happens to coincide with the denominator of the W eyl character formula for sl n under the formal iden tiﬁcation of m i with the i th simple ro ot (1 ≤ i ≤ n − 1). It follo ws from ( 162 ) and ( 163a ) that Z l, m ( x 1 , x 2 , . . . , x L ; t ) is a homo gene ous symmetric p olynomial in x 1 , . . . , x L of degree P n i =1 im i . Theorem 35 together with ( 163a ), implies the factorization Z 1 , m ( x 1 , . . . , x L ; t ) = ∆ m ( t ) − 1 n Y i =1 Z l, ( m 0 + ··· + m i − 1 ,m i + ··· + m n ) ( x 1 , . . . , x L ; t ) . (164) The RHS consists of the partition functions for the single-sp ecies case n = 1, which are readily obtained from Example 30 and the deﬁnition ( 161 ). Example 37. Z 2 , (1 , 1 , 2) ( x 1 , x 2 ; t ) = 1 1 − t Z 2 , (1 , 3) ( x 1 , x 2 ; t ) Z 2 , (2 , 2) ( x 1 , x 2 ; t ) , Z 2 , (1 , 3) ( x 1 , x 2 ; t ) = t (1 + t ) x 1 x 2 ( x 1 + x 2 ) , Z 2 , (2 , 2) ( x 1 , x 2 ; t ) = t ( x 2 1 + x 2 2 ) + (1 + t ) 2 x 1 x 2 , Z 2 , (1 , 2 , 3) ( x 1 , x 2 , x 3 ; t ) = 1 1 − t 2 Z 2 , (3 , 3) ( x 1 , x 2 , x 3 ; t ) Z 2 , (1 , 5) ( x 1 , x 2 , x 3 ; t ) , Z 2 , (3 , 3) ( x 1 , x 2 , x 3 ; t ) = (1 + t )  (1 + t 2 ) x 1 x 2 x 3 + t ( x 1 + x 2 )( x 1 + x 3 )( x 2 + x 3 )  , Z 2 , (1 , 5) ( x 1 , x 2 , x 3 ; t ) = t 2 (1 + t ) x 1 x 2 x 3 ( x 1 x 2 + x 1 x 3 + x 2 x 3 ) . Giv en a lo cal state σ = ( σ 0 , . . . , σ n ) ∈ B l in the m ultiplicit y represen tation ( 6 ), we write b σ = ( σ n , . . . , σ 0 ), whic h corresp onds to the conjugation of particle sp ecies a ↔ n − a . Theorem 35 together with ( 163a )–( 163b ) implies the following symmetry of the partition function under particle conjugation. Corollary 38. Z l, m ( x 1 , . . . , x L ; t ) = ( − 1) n ( n − 1) / 2 t γ ( x 1 · · · x L ) nl Z l, b m ( x − 1 1 , . . . , x − 1 L ; t − 1 ) , (165) wher e b m = ( m n , . . . , m 0 ) for m = ( m 0 , . . . , m n ) , and γ = Lnl ( l − 1) / 2 − P 0 b ) t i b +1 + ··· + i n 1 − t k z . (175) In the general case, an approach based on the three-dimensional integrabilit y and the tetrahedron equation leads to the formula S k,l ( z ) a , b i , j = A ( q l − k /z ) a , b i , j   q = t 1 / 2 , where the function A ( z ) a , b i , j for a , i ∈ B k and b , j ∈ B l is deﬁned by [ K22 , eqs. (13.47)–(13.50), (13.127)] with ( l , m ) there replaced with ( k , l ). F rom the natural identiﬁcation V 1 ≃ V 1 and B 1 = B 1 in ( 5 ) and ( 6 ), the R -matrices S k l ( z ) ∈ End( V k ⊗ V l ) in tro duced in Section 3 and S k,l ( z ) ∈ End( V k ⊗ V l ) are essentially identical in the sp ecial case k = 1. Indeed, they are related by S 1 l  z  = S 1 ,l  t l − 1 z  (1 − t l z ) , (176) whic h coincides with the numerator of ( 174 ). F or l = 1, this follo ws by comparing Example 8 with ( 115 ). Remark 42. As a consequence of the symmetric fusion, the sum ( 172 ) is actually inv ariant under the replacemen t of ( i 1 , i 2 , i 3 ) with any ( i ′ 1 , i ′ 2 , i ′ 3 ) ∈ C ( i ), and likewise of ( j 1 , j 2 ) with any ( j ′ 1 , j ′ 2 ) ∈ C ( j ). A similar in v ariance is v alid for general k and l . MUL TISPECIES t -PUSHT ASEP 39 A cknowledgments A. A. was partially supp orted b y SERB Core grant CRG/2021/001592 as w ell as the DST FIST pro- gram - 2021 [TPN - 700661]. A. K. thanks the organizers of the Pr o gr am on Classic al, Quantum and Pr ob abilistic Inte gr able Systems (March 24–Ma y 24, 2025) at the Center of Mathematical Sciences and Appli- cations, Harv ard Universit y , and the Symp osium on Solvable L attic e Mo dels & Inter acting Particle Systems (August 24–30, 2025) at Schloss Elmau, Germany , for their kind invitation and warm hospitality , where part of this work was carried out. A. K. also thanks Amol Aggarwal, Sylvie Corteel, Leonid Petro v, Michael Wheeler, and Lauren Williams for their interest and useful communications. A. K. is supported b y JSPS KAKENHI No. 24K06882. References [ANP23] A. Aggarwal, M. Nicoletti, L. Petrov, Colored interacting particle systems on the ring: Stationary measures from Y ang-Baxter equation. Compos. Math. 161 no. 8 1855–1922 (2025). [AK25] A. Ayyer, A. Kuniba, Multisp ecies inhomogeneous t -pushT ASEP from antisymmetric fusion, Electron. J. Probab. 30 article no. 190, 1–28 (2025). [AM23] A. 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Sklyanin, L. A. T akhata jan, L. D. F addeev, Quan tum inv erse problem metho d I, Theor. Math. Ph ys. 40 688–706 (1980). Ar vind A yyer, Dep ar tment of Ma thema tics, Indian Institute of Science, Bangalore 560012, India Email address : arvind@iisc.ac.in A tsuo Kuniba, Gradua te School of Ar ts and Sciences, University of Tokyo, Komaba, Tokyo, 153-8902, Jap an Email address : atsuo.s.kuniba@gmail.com

Multispecies inhomogeneous $t$-PushTASEP with general capacity

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