Simplices and spectra of graphs

SIMPLICES AND SPECTRA OF GRAPHS IGOR RIVIN A bstract . In this note we show the n − 2-d imensional volumes of codimension 2 faces of a n n -d imensional simplex are a lgebraically independe nt functions of the lengths of edges. In order to prove this we compute the complete spectrum of a combinatorially interestin g gra ph. I ntroduction Let T n be the set of congruence classes of simplices in Euclidean space E n . The set T n is an open ma nifold (also a semi-algebraic set) of dime n sion ( n + 1) n / 2 . Coincidentally , a simplex T ∈ T n is determined by the ( n + 1) n / 2 le ngths of its edges. Furthermor e, the squar e of the volume of T ∈ T n is a polyno mia l in the squar es of the edgelengths. This polynomial is given by the Cayley-Menger determinant formula: V 2 ( T ) = ( − 1) n + 1 2 n ( j !) n det C , wher e C is the Cayley -Menger matrix, defined as follows: C i j =            0 , i = j 1 , if i = 1 or j = 1, but not both l 2 ( i − 1)( j − 1) , otherwise Note that an n -dime nsional simplex also has ( n + 1) n n − 2 dimensional faces, and so the following q uestion is natural: Question 1 . Is the congr uence class of the simplex T determined by the n − 2- dimensional volumes of the n − 2-dimensional faces? Question 1 must be classical, but the first refer ence that I am aware of is W arr en Smith’s Princeton PhD thesis [5]. Date : October 22, 2021. The author would like to thank the America n Institute of Ma thematics for an invitation to the workshop on “Rigidity and Polyhedral Combinatorics”, wher e this work was started. The a uthor has profited f rom discussions with Igor Pak, Ezra Miller , and B ob Connelly . 1 2 IGOR RIVIN In the AIM workshop on Rigidity and Polyhedral Combinatorics Bob Connelly (who wa s unaware of the refer ence [5]) raised the following: Question 2 . Is t he volume of the simplex T determined by the n − 2-dime nsional volumes of the n − 2-dime nsional faces? In fact, Connelly stat e d Quest ion 2 for n = 4 , which is the first case where the question is open (for n = 3 the answer is trivially ”Y es”, since 3 − 2 = 1 , and we ar e simply asking if the volume of the simple x is determined by its edge-lengths. In dimension 2 , the answer is trivially ”No”, since 2 − 2 = 0 , and the volume of codimension-2 faces of a triangle carries no information. Clearly , the a ffi rmative answer to Question 1 implies an a ffi rmative answer to Question 2. At the time of this writing, bot h questions ar e open. In this note we show Theorem 3. The ( n + 1) n / 2 n − 2 -dimensional volumes of the n − 2 -dimensional faces of an n-dim ensional simpl e x ar e algebr ai c ally independent over C [ l 12 , . . . , l (( n + 1)( n + 1 ] . Theor em 3 is clearly a necessary step in the dir e ction of reso lving Question 1, but is far fr om su ffi cient. T o show it, consider the map of R ( n + !) n / 2 to R ( n + 1) n / 2 , which sends the vector E of e dge-lengths to the vector F of areas of n − 2 -dimensional faces. T o show Theorem 3 it is enough to check that the Jacobian J ( E ) = ∂ F /∂ E is non- singular at one point. W e will use the most obvious point p 1 : one corresponding to a regu la r simplex with all edge-lengths equal to 1 . By symmetry consideratio ns, the Jacobian J ( p 1 ) can be written as J ( p 1 ) = cM , wher e M i j = ( 1 , if the j -th edge is incident to the i -th face of dimension n − 2 . 0 , otherwise The first observation is Lemma 4 . The c onstant c above is not equal to 0 . Proo f . This follows from the observation that the volume of a k -dimensional sim- plex is a homogeneous function of the edge-lengths, of degr ee k . A n application of Euler ’s Homogeneous Function Theorem shows that at p 1 , ∂ F i ∂ e j = ( 2 n − 1 F , if e j is incident to F i 0 , otherwise , wher e F is the co mmon value of the n − 2-dimensional volume of the F i , which implies that c = 2 n − 1 F .  SIMPLICES AND SPECTRA OF GRAPHS 3 Theor em 3 thus reduces to the assertion that the d e terminant of the matrix M is not zer o. W e will be able to pr ove a much st ronger r e sult (of int e rest in its own right): Theorem 5. The s ingular values of M are as follows: Th e value 1 appears with multipli c ity ( n + 1)( n − 1 ) / 2 . The value ( n − 2 ) appears with m ultiplicity n . The value ( n − 2)( n − 1) appears once. Corollary 6. The absolute value of the determ inant of M equals ( n − 2) n + 1 ( n − 1) , 0 , for n > 2 . T o pr ove Theorem 5, firs t r ecall that the sing ular values of M ar e the posit ive squar e ro ots of the e igenvalues of N = MM t . In its turn, N has rows and columns indexed by n − 1-subsets of R n + 1 = { 1 , . . . , n + 1 } . The i j -th element of N equals the number of 2-element subsets the i -th and the j -th two eleme nt subsets s i and s j have in common. This, in turn, can be written a s follows: N i j =            n ( n − 1) / 2 , i = j ( n − 1)( n − 2) / 2 , | s i ∩ s j | = n − 2 ( n − 2)( n − 3) / 2 , | s i ∩ s j | = n − 3 . The matrix N is the adjacency matrix of a multi-graph G N , which has a rather lar ge symmetry gr oup. These symmetr ie s will allow us to obtain the complete spectral de composition of the ma trix N . Indeed, the symmetry gro up of G N is the symmetric group S n + 1 , while the stabilizer Γ i of a vertex s i is the group S n − 1 × S 2 (the first factor acts on s i itself, the second on R n + 1 \ s i . ) The action of Γ i on G N has thr ee or bits. The first consists of s i itself. The second consists of all s j such that | s i ∩ s j | = n − 2 , the third of all s j such that | s i ∩ s j | = n − 3 . At this point it behooves us to recall the concept of graph divisor . 1. G raph divisors The concept of graph divisor is discussed at great length in the books [2] and [1] Definition 7. Give n an s × s m atrix B = ( b i j ) , let the vertex set of a (multi)graph G be partitioned into (non-empty) subsets X 1 , X 2 , . . . , X s , so that for any i , j = 1 , 2 , . . . , s , eac h vertex f rom X i is adjacent to exac tly b i j vertices of X j . The multidi graph H with adjacency matrix B is called a fro nt divisor of G . The importance of this concept to our needs is that the charact eristic polynomial of a graph divisor divides the characteristic polynomial of the adjacency matrix of G . (hence the name ). The most interesting (to us, anyway) example of a graph d ivisor 4 IGOR RIVIN arises by having a subgr oup Γ of the automorphism gro up of G . The quotient of G by Γ is the divisor we consider . Every eigenvector of Γ \ G lifts to an eigvenvector of G with the same eigenvalue. If we consider our graph G N and the action of Γ i , we observe (aft e r a rather tedious computation) that the fr ont divisor corresponding to the act ion of Γ i on G N has adjacency matrix D =           1 2 ( n − 2)( n − 1) ( n − 3)( n − 2)( n − 1) 1 4 ( n − 4)( n − 3)( n − 2)( n − 1) 1 2 ( n − 3)( n − 2) n 3 − 7 n 2 + 17 n − 14 1 4  n 4 − 10 n 3 + 39 n 2 − 70 n + 48  1 2 ( n − 4)( n − 3) n 3 − 8 n 2 + 23 n − 24 1 4  n 4 − 10 n 3 + 43 n 2 − 90 n + 76            A simple computatio n shows that the eigenvalues of D a re ( n − 1) 2 ( n − 2) 2 , ( n − 2) 2 , 1 , while the corr esponding eigenvectors are (respect ively): (1 , 1 , 1) , ((1 − n ) / 2 , (3 − n ) / 4 , 1 ) , ((2 − 3 n + n 2 ) . 2 , (2 − n ) / 2 , 1 ) . Graph divisors are a very useful tool, but they have too obvious shortcomings : (1) Not all eigenvalues of the graph G are captured by the graph divisor . (2) W e have no information on the multiplicity of any of t he eigenvalues that ar e captured. Here, however , we have a deus ex machina in the form of 2. G elf and p airs W e will not need any more than the (rather little) pr esented in Diaco nis’ little book [3, page 54] First: Definition 8. Let G be a group acting transitively on a finite set X with isotropy grou p N . A function f : G → C is called N -bi-invariant if f ( n 1 gn 2 ) = f ( g ) , for all n 1 , n 2 ∈ N , g ∈ G . The pai r G , N is call ed a Gelfand pair i f the convolution of N -bi-invariant functions is commutative. In our application, G = S n + 1 , N = S n − 1 × S 2 , and X = G N . The further r e sults we need are the following (the citations are from [3]: Lemma 9 (Lemma 5, page 53) . Let ρ, V be an irr ed ucible repr esentation of the finite grou p G . Let N ⊂ G be a subgrou p and let X = G / N be the ass ociated homoge neous space. The number of times that ρ appears i n L 2 ( X ) e quals the d imension of the space of N -fixed vectors in ρ, V . SIMPLICES AND SPECTRA OF GRAPHS 5 Theorem 1 0 (Theor e m 9, page 54) . Th e foll owing three conditions are equivale nt: (1) G , N is a Gelfand pair . (2) The decomposi tion of L 2 ( X ) into irr e ducible r epresentations of G is multiplicity- free. (3) For every irreducible representation ( ρ, V ) there is basis of V such that ˆ f ( ρ ) =  ∗ 0 0 0  , for all N -bi-invari ant functions f . The main significance (as seen by this author) of the Gelfand pair pr operty is that the eigenvalues of N \ G / N (which is the ”front divisor” D ) are the same as the eigenvalues of X = G / N , since e ach irreducible factor of L 2 ( X ) cont ains an invariant function (as per Lemma 9) W e will finally need the following (attributed by Diaconis to [4]): Fact 11. The pair S n + 1 , S n − 1 × S 2 is a Gelfand pair . The dimensions of the irreducible repr esentations of G n are ( n + 1) n / 2 , n , 1 . W e now have e verything we need to finish the proof of Theorem 5, and hence Theor em 3. Since the graph laplacian (or adjacency matrix) commutes with the action of the automorphism gr oup, each eigenspace of the adjacency matrix M is a sum of irr e ducible repr esentations of S n + 1 . By Fact 11 there ar e precisely thr e e eigenspaces, and their dimensions are as stated in Theorem 5. The only question is how to decide which eigenspace has which dimension. Since we know the eigenvectors of the “front divisor ” matrix D (see the end of the I ntr oduction) we can match them up with the “spherical funct ions” of the Gelfand pair S n + 1 , S n − 1 × S 2 (as given on [3, p a ge 57], whence the result follows. R eferences [1] D. Cvetkovi ´ c, P . Rowlinson, and S. Simi ´ c. Eigenspaces of graphs , volume 6 6 of Encyclopedia of Mathematics and its Applications . Cambridge University Pr e ss, Cambridge, 1997. [2] Drago ˇ s M. C v e tkovi ´ c, Michael Doob, and Horst Sachs. Spectra of graphs . Johann Ambrosius Barth, Heidelberg, thir d edition, 199 5 . Theory and app lications. [3] Persi Diaconis. Group representations in p robability and statistics . Institute of Mathematical Statis- tics Lecture Notes—Monograph Serie s, 11 . Institute of Ma thematical Statistics, Hayward, CA, 1988. [4] G. D. James. The repres entat io n theory of the symmetric gr oups , volume 682 of Lectur e Notes in Mathematics . Springer , B erlin, 1978. [5] W arren Douglas Smitth. Studies in comp utational geometry motivated by mesh generation . PhD thesis, Princeton University , 19 89. D ep artment of M a thema tics , T emple U niversity , P hiladelphia E-mail address : ri vin@t emple. edu