SIMPLICIAL MATRIX-TREE THEOREMS

Transcription

1 SIMPLICIAL MATRIX-TREE THEOREMS ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN Abstract. We generalize the definition of a spanning tree from graphs to arbitrary simplicial complexes, following G. Kalai. We show that the simplicial spanning trees of a complex can be counted, weighted by the squares of the orders of their top-dimensional integral homology groups, using its Laplacian matrix. A more refined count of trees can be obtained by assigning a variable to each vertex of and replacing the Laplacian with a monomial matrix. These results generalize the classical matrix-tree theorem, and its weighted analogue, from graphs to complexes. When is a shifted complex, we give a complete description of the eigenvalues of the weighted Laplacian in terms of the shifted order on its faces, generalizing previously known results for threshold graphs and for unweighted Laplacian eigenvalues. Contents 1. Introduction 2 2. Notation and definitions Simplicial complexes Multisets Combinatorial Laplacians 2 3. Simplicial spanning trees 3 4. Simplicial analogues of the matrix-tree theorem 5 5. Weighted enumeration of simplicial spanning trees 8 6. Shifted complexes and the start of) counting their spanning trees 9 7. The vertex weighting Vertex-weighting, cones, and near-cones Boundary and coboundary operators of cones Eigenvectors of cones The vertex weighting and near-cones z-polynomials and critical pairs z-polynomials Critical pairs The theorem, and its consequences An example: the bipyramid Return to spanning trees of shifted complexes Corollaries Fine enumerator from shifted to threshold Fine enumerator from threshold to Ferrers 28 References 29 Date: July 15,

2 2 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN 1. Introduction 2. Notation and definitions 2.1. Simplicial complexes. Let V be a finite set. A simplicial complex on V is a family of subsets of V such that 1) ; 2) If F and G F, then G. The elements of V are called vertices of, and the faces that are maximal under inclusion are called facets. The dimension of a face F is dim F = F 1, and the dimension of is the maximum dimension of a face or facet). The abbreviation d indicates that dim = d. We say that is pure if all facets have the same dimension; in this case, a ridge is a face of dimension dim 1. Let i be the set of i-dimensional faces of, and let f i ) = i. The i-dimensional skeleton of is the subcomplex i) = j. 1 j i The pure i-dimensional skeleton of is the subcomplex [i] generated by the i-dimensional faces, that is, [i] = F : F G, for some G i }. We assume that the reader is familiar with simplicial homology see, e.g., [13, 2.1]). Let d be a simplicial complex and 1 i d. Let R be a ring if unspecified, assumed to be Z), and let C i ) be the i th simplicial chain group, i.e., the free R-module with basis [σ]: σ i }. We define the simplicial boundary and coboundary maps respectively by,i : C i ) C i 1 ),,i : C i 1 ) C i ) where we have identified cochains with chains via the natural inner product). We will abbreviate the subscripts in the notation for boundaries and coboundaries whenever no ambiguity can arise. We will often regard i resp. i ) as a matrix whose columns and rows resp. rows and columns) are indexed by i and i 1 respectively. The i th reduced) homology group of is H i ) = ker i )/ im i+1 ), and the i th reduced) Betti number β i ) is the rank of the largest free R-module summand of H i ) Multisets. We will often work with multisets of eigenvalues or of vertices), in which each element may occur with positive integer multiplicity. For brevity, we drop curly braces and commas when working with multisets of integers: for instance, 5553 denotes the multiset in which 5 occurs with multiplicity three and 3 occurs with multiplicity one. The cardinality of a multiset is the sum of the multiplicities of its elements; thus 5553 = 4. We write a = b to mean that the multisets a and b differ only in their respective multiplicities of zero; for instance, 5553 = = Of course, = is an equivalence relation.) The union operation on multisets is understood to add multiplicities: for instance, = Combinatorial Laplacians. We adopt the notation of [9] for the Laplacian operators or matrices) of a simplicial complex. We summarize the notation and mention some fundamental identities here. For 1 i dim, define linear operators L ud,i, Ldu,i, Ltot,i on the vector space C i ) by L ud,i = i+1 i+1 L du,i = i i L tot,i = L ud,i + L du,i the up-down Laplacian), the down-up Laplacian), the total Laplacian).

3 SIMPLICIAL MATRIX-TREE THEOREMS 3 Let s tot i ) be the spectrum of L tot,i ; that is, the multiset of its eigenvalues including zero). Similarly, let i ) and s du i ) be the spectra of L ud,i and Ldu,i respectively. Since each Laplacian operator is represented by a symmetric matrix, it is diagonalizable, so s tot i ) = i ) = s du ) = f i ). The various Laplacian spectra are related by the identities i ) = s du i+1 ), s tot i ) = i i ) s du ) [9, eqn. 3.6)]. Therefore, each of the three families of multisets s tot i ): 1 i dim }, i ): 1 i dim }, s du i ): 1 i dim } determines the other two, and we will feel free to work with whichever one is most convenient in context. i 3. Simplicial spanning trees Definition 3.1. Let be a d-dimensional simplicial complex, and let k d. A k-dimensional simplicial spanning tree for short, SST or k-sst) of is a k-dimensional subcomplex Υ such that Υ k 1) = k 1) and H k Υ) = 0, 1a) H k 1 Υ) <, and 1b) f k Υ) = f k ) β k ) + β k 1 ). 1c) We write T k ) for the set of all k-ssts of, omitting the subscript if k = d. T k ) = T j) ) for all j k. Note that An 0-dimensional spanning tree is just a vertex of. If is a 1-dimensional simplicial complex on n vertices that is, a graph then the definition of 1-SST coincides with the usual definition of a spanning tree of a graph. Namely, a subgraph of which is connected, acyclic, and has n 1 edges. Next, we give a few examples in higher dimensions. Example 3.2. If is a simplicial sphere for instance, the boundary of a simplicial polytope), then deleting any facet of produces a simplicial spanning tree. Hence a k-dimensional simplicial sphere has f k ) SSTs. Example 3.3. In dimension > 1, spanning trees need not be acyclic. For example, let be a triangulation of the real projective plane, so that dim = 2 and H 2 ) = Z/2Z; then is a 2-SST of itself. Example 3.4. Consider the two-dimensional complex with facets 123, 124, 125, 134, 135, 234, 235}, the bipyramid with equator. There are two ways to obtain a SST of this complex; remove the equator facet 123 and any one boundary facet or remove one boundary facet from each of the two pyramids of the bipyramid. Thus the number of SSTs is 15: six of the first type and nine of the second. The next fact generalizes the fundamental property of spanning trees of graphs that any two of the three conditions connected, acyclic, has n 1 edges implies the third. Proposition 3.5. Let Υ be a subcomplex with Υ k 1) = k 1). Then any two of the conditions 1a), 1b), 1c) imply the third.

4 4 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN Proof. Note that f l Υ) = f l ) for l k 1, β l Υ) = β l ) for l k 2. 2) We calculate the Euler characteristic χυ) in two ways, as the alternating sum of the f-numbers and of the Betti numbers. First, On the other hand χυ) = χυ) = k 1) i f i Υ) i=0 k 1) i β i Υ) i=0 Equating 3) and 4) gives or equivalently k 1 = 1) k f k Υ) + 1) i f i ) i=0 = 1) k f k Υ) + χ ) 1) k f k ). 3) k 2 = 1) k β k Υ) β k 1 Υ)) + 1) i β i ) i=0 = 1) k β k Υ) β k 1 Υ)) + χ ) 1) k β k ) β k 1 )). 4) f k Υ) f k ) = β k Υ) β k 1 Υ) β k ) + β k 1 ) fk Υ) f k ) + β k ) β k 1 ) ) β k Υ) + β k 1 Υ) = 0. Since 1a) says that β k Υ) = 0 note that H k Υ) must be free abelian) and 1b) says that β k 1 Υ) = 0, the conclusion follows. Definition 3.6. A simplicial complex is called metaconnected if β j ) = 0 for all j < dim. Equivalently, a d-dimensional complex is metaconnected if it has the homology type of a wedge of zero or more d-dimensional spheres. In particular, any Cohen-Macaulay complex is metaconnected. The converse is very far from true, because, for instance, a metaconnected complex need not even be pure. For our purposes, metaconnected complexes are the right simplicial analogues of connected graphs for the reason stated in the next proposition. Proposition 3.7. A simplicial complex has a simplicial spanning tree if and only if it is metaconnected. Proof. Let d = dim. Suppose that β k ) = 0 for all k < d. Then for each k < d, we can construct a spanning k-tree Υ by taking the entire k-skeleton of and removing facets until we get a complex whose k-dimensional homology vanishes. On the other hand, suppose that k < d and that Υ is a simplicial spanning k + 1)-tree of. Consider the following diagram: C k+1 ) C k+1 Υ),k Υ,k Ck ) Ck Υ),k 1 Υ,k 1 Ck 1 ) Ck 1 Υ) Then ker,k 1 = ker Υ,k 1 and im,k im Υ,k, so there is a surjection H k Υ) H k ). The domain is finite by definition of Υ, so the target is also finite.

5 SIMPLICIAL MATRIX-TREE THEOREMS 5 Metaconnectedness is a fairly mild condition on simplicial complexes. Any Z-acyclic complex is metaconnected and is its own unique spanning tree), as is any Cohen-Macaulay complex. In particular, shifted and matroid complexes, which we will study in more detail later, are metaconnected. 4. Simplicial analogues of the matrix-tree theorem Throughout this section, let d be a metaconnected simplicial complex. For k d, define π k = π k ) = λ, h k = h k ) = H k 1 Υ) 2. 0 λ k 1 ) Υ T k ) We are interested in the relationships between these two families of invariants. When d = 1, the relationship is given by the classical matrix-tree theorem; see, e.g., [17, 20]. Theorem Matrix-Tree Theorem I product of nonzero eigenvalues). Let be a graph i.e., a 1-dimensional simplicial complex) on n vertices. Then h 1 = π 1 n. The matrix-tree theorem is often reformulated in terms of the reduced Laplacian. Let =,1 be the simplicial boundary matrix of equivalently, the signed vertex-edge incidence matrix). Deleting the row of corresponding to a vertex i produces the reduced boundary matrix i. The reduced Laplacian is then L i = i i ; this is an n 1) n 1) symmetric matrix. Theorem Matrix-Tree Theorem II determinant of reduced Laplacian). With the notation above, we have for all i h 1 = det L i. The goal of this section is to generalize both versions of the matrix-tree theorem to arbitrary metaconnected complexes d. Our results extend the work of Kalai [15] and Adin [1] on simplicial generalizations of complete graphs and complete bipartite graphs respectively. Indeed, Kalai s and Adin s arguments can be generalized to all metaconnected complexes, and we proceed by mimicking their approach. Throughout the rest of the section, abbreviate β i = β i ), f i = f i ), and =,d. Let T be a set of facets of of cardinality f d β d + β d 1 = f d β d, and let S be a set of ridges such that S = T. Define T = T d 1), S = d 1) \ S, S = S d 2), =,d. Also, let S,T be the square submatrix of with rows indexed by S and columns indexed by T. Proposition 4.1. The matrix S,T is nonsingular if and only if T T d ) and S T d 1 ). Proof. We may regard S,T as the top boundary map of the d-dimensional relative complex Γ = T, S). So S,T is nonsingular if and only if H d Γ) = 0. Consider the long exact sequence 0 H d S) H d T ) H d Γ) H d 1 S) H d 1 T ) H d 1 Γ) 5) If H d Γ) 0, then H d T ) and H d 1 S) cannot both be zero. This proves the only if direction. If H d Γ) = 0, then H d S) = 0 since dim S = d 1), so 5) implies H d T ) = 0. Therefore T is a d-tree, because it has the right number of facets. Hence H d 1 T ) is finite. Then 5) implies that H d 1 S) is finite. In fact, it is zero because the top homology group of any complex must be torsion-free. Meanwhile, S has the right number of facets to be a simplicial spanning d 1)-tree of, proving the if direction. Proposition 4.2. If S,T is nonsingular, then det S,T = H d 1 T ) H d 2 S) H d 2 T ) = H d 1 T ) H d 2 S) H. d 2 )

6 6 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN Proof. As before, we interpret S,T as the boundary map of the relative complex Γ = T, S). So S,T is a map from Z T to Z T, and Z T / S,T Z T ) is a finite abelian group of order det S,T. On the other hand, since Γ has no faces of dimension d 2, its lower boundary maps are all zero, so det S,T = H d 1 Γ). Since H d 2 T ) is finite, the desired result now follows from the piece 0 H d 1 T ) H d 1 Γ) H d 2 S) H d 2 T ) 0 6) of the long exact sequence 5). Theorem 4.3 Simplicial Matrix-Tree Theorem I product of non-zero eigenvalues). Let be a metaconnected complex of dimension d. Then π d ) = h d )h d 1 ) H d 2 ) 2. Proof. Let L = L ud,d 1. This is a square matrix whose size is f d 1 and whose rank is f d β d = f d β d + β d 1 by metaconnectedness). Let χl; y) = detyi L) be its characteristic polynomial, so that π d ), the product of the nonzero eigenvalues of L, is given up to sign) by the coefficient of y f d 1 f d +β d in χl; y). Equivalently, π d = det L S = det L S, 7) S d 1 S =rank L S d 1 S =f d β d where L S denotes the square submatrix of L with columns and rows indexed by S. By the Binet- Cauchy formula, we have det L S = det T,S)det S,T ) = det S,T ) 2. 8) T d T = S T d T = S Combining 7) and 8), applying Proposition 4.1, and interchanging the sums, we obtain π d = det S,T ) 2 and now applying Proposition 4.2 yields π d = T T d ) S T d 1 ) T T d ) S T d 1 ) H d 1 T ) H d 2 S) H d 2 ) ) 2 as desired. = H d 1 T ) 2 H d 2 S) T T d ) S T d 1 ) H d 2 ) 2 We next seek a simplicial analogue of the second version of the matrix-tree theorem. First, we check that when we delete the rows of corresponding to a simplicial spanning d 1)-tree, the resulting reduced Laplacian has the correct size, namely, that of a spanning d-tree. Lemma 4.4. Let U T d 1 ) and S = d 1 \U. Then S = f d ) β d ), the number of facets of a spanning tree of.

7 SIMPLICIAL MATRIX-TREE THEOREMS 7 Proof. Let Γ = d 1). By Proposition 3.5 and the observation 2), U = f d 1 Γ) β d 1 Γ) + β d 2 Γ) = f d 1 ) β d 1 Γ), so S = β d 1 Γ).. The Euler characteristics of and Γ are By 2), we see that d d χ ) = 1) i f i ) = 1) i β i ), i=0 i=0 d 1 d 1 χγ) = 1) i f i Γ) = 1) i β i Γ). i=0 i=0 χ ) χγ) = 1) d f d ) = 1) d β d ) + 1) d 1 β d 1 ) 1) d 1 β d 1 Γ) from which we obtain f d ) = β d ) β d 1 ) + β d 1 Γ). Since is metaconnected, we have β d 1 ) = 0, so S = β d 1 Γ) = f d ) β d ) as desired. Theorem 4.5 Simplicial Matrix Tree Theorem II determinant of reduced Laplacian). Let U T d 1 ) and S = d 1 \U. Let S be the reduced boundary matrix obtained from by deleting the rows corresponding to ridges in U, and let L S = S S. Then Proof. The Binet-Cauchy formula gives h d ) = H d 2 ) 2 H d 2 U ) 2 det L S. det L S = det S S) = T : T = S = T : T = S det S,T )det S,T ) det S,T ) 2. By Lemma 4.4 and Proposition 4.1, S,T is nonsingular exactly when T corresponds to a simplicial spanning tree of. Hence Proposition 4.2 gives det L S = H d 1 T ) H ) 2 d 2 U ) H d 2 ) T T d ) = H d 2 U ) 2 H d 2 ) 2 T T d ) H d 1 T ) 2 = H d 2 U ) 2 H d 2 ) 2 h d ), which is equivalent to the desired formula. Remark 4.6. Suppose that H d 2 ) = 0. This occurs, for example, if is Cohen-Macaulay.) Then the two versions of the theorem assert that h d = π d det L S = h d 1 H d 2 U ), 2 from which it is easy to recognize the two different versions of the classical matrix-tree theorem for graphs. Note that a graph is Cohen-Macaulay as a simplicial complex if and only if it is connected.) Moreover, the recurrence h d = π d /h d 1 leads to an expression for h d as an alternating product of eigenvalues: h d = π dπ d 2 π d 1 π d 3.

8 8 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN 5. Weighted enumeration of simplicial spanning trees We can obtain much finer enumerative information by labeling the facets of a complex with indeterminates, so that the invariants h i and π i become generating functions for its simplicial spanning trees. The proofs of Theorems 4.3 and 4.5 carry over easily to this more general setting. As before, let d be a pure metaconnected simplicial complex, and let =,d. Introduce an indeterminate x F for each facet F, and let X F = x 2 F. For every T d, let x T = F T x F and let X T = x 2 T. To construct the weighted boundary matrix ˆ from, multiply each column of ˆ by x F, where F is the facet of corresponding to that column. The weighted coboundary ˆ is the transpose of ˆ. We can now define weighted versions of Laplacians, the various submatrices of the boundary and coboundary matrices used in Section 4, and the invariants π k and h k, denoting each by placing a hat over the symbol for its unweighted analogue. Thus ˆπ k is the product of the nonzero ud eigenvalues of ˆL,k 1, and ĥ k = ĥk ) = H k 1 Υ) 2 X Υ. Υ T k ) To recover any unweighted quantity from its weighted analogue, set x F = 1 for all F d. Lemma 5.1. Let T d and S d 1 such that T = S = f d β d. Then det ˆ S,T = x T det S,T is non-zero if and only if T T d ) and S T d 1 ). In that case, ± det ˆ S,T = H d 1 T ) H d 2 S) H d 2 T ) x T = H d 1 T ) H d 2 S) H x T. 9) d 2 ) Proof. The first claim follows from Proposition 4.1, and the second from Proposition 4.2. Theorem 5.2 Weighted Simplicial Matrix-Tree Theorem I product of non-zero eigenvalues). Let be a d-dimensional, pure, metaconnected simplicial complex. Then 1 ˆπ d ) = ĥd )h d 1 ) H d 2 ) 2. Proof. The argument is essentially identical to that of Theorem 4.3. Let ˆL = the Binet-Cauchy formula and Lemma 5.1, we have ˆπ d = det ˆL S = det ˆ T,S)det ˆ S,T ) = det ˆ S,T ) 2 S S T S d 1 S =f d β d = T T d ) = T T d ) S T d 1 ) S T d 1 ) T d T = S det ˆ S,T ) 2 ud ˆL,d 1 ; then, using H d 1 T ) H ) 2 d 2 S) H X T = ĥd )h d 1 ) d 2 ) H. d 2 ) 2 Theorem 5.3 Weighted Simplicial Matrix Tree Theorem II determinant of reduced Laplacian). Let be a d-dimensional, pure, metaconnected simplicial complex. Let U T d 1 ), let S = Ū, and let ˆL ud be the reduced Laplacian ˆL S. Then ĥ d ) = H d 2 ) 2 H det ˆL. d 2 U ) 2 1 There are no missing hats on the right-hand side of this formula! Only hd ) has been replaced with its weighted analogue, not h d 1 ).

9 SIMPLICIAL MATRIX-TREE THEOREMS 9 Proof. The argument is essentially identical to that of Theorem 4.5, using Lemma 5.1 instead of Proposition Shifted complexes and the start of) counting their spanning trees [Note: This used to be near the end of the paper. There should be some transition words saying how now we re going to apply what we ve done so far, to shifted complexes] A complex Σ on vertex set V is called a near-cone with apex p if it has the following property: if σ del p Σ and v σ, then σ\v link p Σ equivalently, σ\v p Σ). [Note: Perhaps somewhere define family.] If A = a 1 < a 2 < < a k } and B = b 1 < b 2 < < b k } are two k-sets of integers, then A < B in the componentwise partial order if a j b j for all j. A k-family F is shifted if B F and A < B together imply A F; equivalently, F is shifted if it is an order ideal with respect to <. A simplicial complex Σ is shifted if Σ i is shifted for all i, with the same ordering of vertices for each i. [Give references for all this?] We will not lose any generality in what we do by assuming that the vertex set for every shifted complex we encounter is of the form [p, q] := p, p + 1,..., q}. It is easy to see that, if a complex is shifted on vertex set [p, q], then its deletion and link with respect to vertex p are again each shifted on vertex set [p + 1, q]. 2 It is also easy to see that a shifted complex on with initial vertex p is a near-cone with apex p. Björner and Kalai [4, Theorem 4.3] showed that the Betti numbers of a shifted complex Σ indeed, of a near-cone) with initial vertex p are given by β i Σ) = F Σ i : p F, F p Σ}. 10) Let Σ be a pure shifted d-dimensional complex with initial vertex p. We would like to use the simplicial matrix-tree theorem to give a weighted enumeration of simplicial spanning trees of Σ using the doubly indexed weighting 13): that is, we wish to compute ĥ d Σ) = H d 1 Σ, Z) 2 X T, where T T Σ) X T = facets σ T Accordingly, define the finely weighted simplicial boundary map of Σ as the homomorphism ˆ = ˆ Σ,i : C i Σ) C i 1 Σ) given by X σ. ˆ [σ] = j σ εj, σ) d i x σ )[σ\j] 11) and similarly define the finely weighted simplicial coboundary map ˆ = ˆ Σ,i+1 : C iσ) C i+1 Σ) as ˆ [σ] = εj, σ j) d i 1 x σ j )[σ j]. 12) j V \σ These maps do not make the chain groups of Σ into an algebraic chain complex, because ˆ ˆ and ˆ ˆ do not vanish in general. However, they are combinatorially significant, because we can apply Version II of the weighted simplicial matrix-tree theorem to the finely weighted up-down Laplacian ˆL ud = ˆ d ˆ d, as follows. Regard this Laplacian as a matrix whose rows and columns are indexed by the ridges of Σ. Let T be the simplicial spanning tree of Σ d 1) consisting of all ridges containing vertex p, and let R be the reduced Laplacian obtained from L by deleting the corresponding rows 2 This is also true for the deletion and link with respect to any vertex, not just p, but then the resulting vertex set is no longer a set of consecutive integers. Since we will not have any need to take the deletion and link with respect to any vertex other than p, we won t worry about that, and instead enjoy the resulting simplicity of specifying the new minimal vertex of the deletion and link.

10 10 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN and columns, so that the remaining rows and columns are indexed by the facets of Λ := link p Σ. Then Version II of the weighted simplicial matrix-tree theorem asserts that ĥ d Σ) = det R. Consider the matrix N obtained from R by dividing each entry in row σ and column τ by the monomial x σ ) x τ ). Then det R = X σ det N. σ link p Σ We now claim that N = X 1,p I + d d, where I is an appropriately sized identity matrix and d, d are the vertex weighted boundary and coboundary maps. Given the claim, it follows that det N = χ M, X 1,p ) = X 1,p + λ), eigenvalues λ of M where χ denotes the characteristic polynomial of M in the variable X 1,p. Note that M = d d = L ud,d 1, where := del p Σ. [Note: The above claim, and all the other missing details of the first part of this section are discussed in reduction.tex, although in a breezier tone than we ll want in the paper. I ll work on rewriting that file to include in here Art] We will spend most of the rest of the paper computing the eigenvalues of M, in order to determine ĥ d Σ). 7. The vertex weighting We now study a particular weighting by Laurent monomials, defined with respect to a total ordering of the vertices of. This weighting allows us to keep track of the individual vertices and the order in which they appear in faces, while maintaining the structure of an algebraic chain complex. It is easy to translate between the vertex weighting and a natural combinatorial weighting of the sort treated in Section 5. Moreover, the vertex weighting behaves well with respect to cones and near-cones, providing a valuable tool in describing the finely weighted Laplacian eigenvalues of shifted complexes and thus in enumerating their simplicial spanning trees. Let x i,j } be a lot of indeterminates. Let k be the field of rational functions in the x i,j with coefficients in C or any other field of characteristic zero). Since these indeterminates will often appear squared, we set X i,j = x 2 i,j. Let be the raising operator, defined by x i,j = x i+1,j for i d and x d+1,j ) = 0, extended linearly and multiplicatively to an operator on all of k. The raising operator can also be applied to a k-linear operator f by the rule f)v ) = f 1 V ))), where V denotes any vector over k. We write a for the a th iterate of. For a multiset of vertices S = v 1 v 2 v m } i, define monomials x S = x 1,v1 x 2,v2 x m,vm, X S = X 1,v1 X 2,v2 X m,vm. We point out some useful identities involving the raising operator. Let S = S 1} where denotes the union as multisets, so that the multiplicity of 1 in S is one more than its multiplicity in S). Then, for all integers p, j, p x 1,1 ) p+1 x S j ) = p x 1,1 x S j ) = p x S j ) 13) 14a) and a x S) a+1 x S ) = x a+1,1 = a x 1,1 ). 14b) The same identities hold if x is replaced with X. Until further notice, let be a pure) d-dimensional simplicial complex on V N. Let i denote the set of i-dimensional faces of. Let C i = C i ) be the i th simplicial chain group, that is, the

11 SIMPLICIAL MATRIX-TREE THEOREMS 11 k-vector space with basis [σ]: σ i }. For a vertex v and a face σ, let εv, σ) = 1) j+1 if v is the j th smallest vertex of σ, and εv, σ) = 0 if v σ. Definition 7.1. The vertex-weighted simplicial boundary map of is the homomorphism,i : C i ) C i 1 ) given by,i [σ] = d i x σ ) εj, σ) d i+1 [σ\j] 15) x j σ σ\j ) and similarly the vertex-weighted simplicial coboundary map,i+1 : C i ) C i+1 ) is given by,i+1[σ] = j V \σ εj, σ j) d i 1 x σ j ) d i [σ j]. 16) x σ ) We will sometimes drop one or both subscripts when stating a general property of these operators. The vertex-weighted boundary and coboundary operators can be represented as matrices with entries in k; for instance,,i has f i 1 ) rows and f i ) columns. Therefore, we can apply the raising operator to and by applying it to each matrix entry. That is, if [σ] C i ) and a N, then a,i [σ] = d i+a x σ ) εj, σ) d i+a+1 [σ\j], 17) x j σ σ\j ) a,i[σ] = j V \σ εj, σ j) d i+a 1 x σ j ) d i+a [σ j]. 18) x σ ) The vertex-weighted boundary and coboundary operators induce chain complexes as we now show. C i+1 ),i+1 C i ),i C i 1 ) C i+1 ),i+1 C i ),i C i 1 ) Lemma 7.2. Let a N. Then a a = 0 and a a = 0. Proof. Since the matrices that represent the maps and are mutual transposes, it suffices to prove the first assertion. For σ i, we have by 17) a,i 1 a,i [σ])) = a,i 1 d i+a x σ ) εj, σ) d i+a+1 x j σ σ\j ) [σ\j] = d i+a x σ ) εj, σ) d i+a+1 x j σ σ\j ) a,i 1 [σ\j]) = d i+a x σ ) εj, σ) d i+a+1 x j σ σ\j ) = j σ = j,k σ j k k σ\j k σ\j εk, σ\j) d i+a+1 x σ\j ) d i+a+2 x σ\j\k ) [σ\j\k] d i+a x σ ) εj, σ)εk, σ\j) d i+a+2 x σ\j\k ) [σ\j\k] εj, σ)εk, σ\j) + εk, σ)εj, σ\k) ) d i+a x σ ) d i+a+2 x σ\j\k ) [σ\j\k] and it is a standard fact from simplicial homology theory that the parenthesized expression is zero.

12 12 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN Note: Much of the folowing is applicable to any old weighted boundary map that satisfies 2 = 0.) We define the vertex-weighted) up-down, down-up, and total Laplacians by L ud,i =,i+1,i+1, L du,i =,i,i, L tot,i = L ud,i + L du,i. Each of these is a linear endomorphism of C i ), and is represented by a symmetric matrix, hence is diagonalizable. Let i ), s du i ), and s tot i ) denote the spectra multisets of eigenvalues) of L ud,i, Ldu,i, and Ltot,i respectively. We will use the abbreviated notation Lud, L du, L tot,, s du, s tot when no confusion can arise. Proposition 7.3. Let 1 i d = dim. 1) The chain group C i ) decomposes as a direct sum C i ) = Ci ud ) Ci du ) Ci 0 ) 19) where Ci ud ) has a basis consisting of eigenvectors of L ud i whose eigenvalues are all nonzero, and on which L du i acts by zero; Ci du ) has a basis consisting of eigenvectors of L du i whose eigenvalues are all nonzero, and on which L ud i acts by zero; L ud i and L du i both act by zero on Ci 0 ). 2) kerl ud,i ) = ker,i) and kerl du i ) = ker,i ). 3) dim Ci 0 ) = β i ), the i th Betti number of. 4) Each of the spectra i, s du i, s tot i has cardinality f i ) as a multiset, and s du i = i 1, and 20a) s tot i = i s du i = i i 1. 20b) Proof. For assertion 1), note that L ud i L du i = = 0 and L du i L ud i = = 0. Thus we may simply take Ci ud ) and Ci du ) to be the spans of the eigenvectors of L ud i and L du i with nonzero eigenvalues. For assertion 2), the operators L ud i and,i act on the same space, namely C i ), and they have the same rank this is just the linear algebra fact that rankmm T ) = rank M for any matrix M). Therefore, their kernels have the same dimension. Clearly kerl ud i ) ker,i), so we must have equality. The same argument shows that kerl du i ) = ker,i ). Assertion 3) is proved as follows: β i ) = dim H i, Q) = dim ker,i ) dim im,i+1 ) = f i rank,i ) rank,i+1 ) = dim C i dim C ud i dim C du i = dim C 0 i. For assertion 4), first note that dim C i ) = f i ), and that the matrix representing each spectrum is symmetric, hence diagonalizable. The identity 20a) is a standard fact in linear algebra, and 20b) is a consequence of the decomposition 19). 8. Vertex-weighting, cones, and near-cones 8.1. Boundary and coboundary operators of cones. Let Γ be the simplicial complex with the single vertex 1, and let be any pure d-dimensional complex on V = [2, n]. For σ, write σ = 1 σ. The corresponding cone is We will make use of the identification Ω = 1 = Γ = σ, σ : σ }. C i Ω) = C 1 Γ) C i )) C 0 Γ) C i 1 )).

13 SIMPLICIAL MATRIX-TREE THEOREMS 13 By 17) and 18), the raised) boundary and coboundary maps on Γ are given explicitly by a Γ [1] = x a+1,1 [ ], a Γ[1] = 0, a Γ [ ] = 0, a Γ[ ] = x a+1,1 [1]. Next, we give explicit formulas for the maps Ω,i and Ω,i+1. How these maps act on a face of Ω depends on whether it is of the form σ, for σ i, or σ, for σ i 1. Note that in any case ε1, σ) = 1, and that for all v σ we have εv, σ) = εv, σ). Therefore Ω,i [ ] [σ]) = εj, σ) d i+1 x σ ) d i+2 [ ] [σ\j] x j σ σ\j ) = d i x σ ) εj, σ) d i+1 [ ] [σ\j] x j σ σ\j ) = id,i )[ ] [σ]), 21a) Ω,i [1] [σ]) = ε1, σ) d i+1 x σ ) d i+2 x σ ) [ ] [σ] + εj, σ) d i+1 x σ ) d i+2 [1] [σ\j] j σ x σ\j ) = x d i+2,1 [ ] [σ] εj, σ) x d i+2,1 d i+2 x σ ) x j σ d i+3,1 d i+3 [1] [σ\j] x σ\j ) = x d i+2,1 [ ] [σ] x d i+2,1 εj, σ) d i+1 x σ ) x d i+3,1 d i+2 [1] [σ\j] x j σ σ\j ) = d i+1 Γ id x ) d i+2,1 id,i 1 [1] [σ]), 21b) x d i+3,1 Ω,i+1[ ] [σ]) = ε1, σ) d i x σ ) d i+1 [1] [σ] + εj, σ j) d i x σ j ) x σ ) d i+1 [ ] [σ j] x σ ) j V \σ = x d i+1,1 [1] [σ]+ εj, σ j) d i 1 x σ j ) d i [ ] [σ j] x σ ) j V \σ ) = d i Γ id + id,i+1 [ ] [σ]), 21c) Ω,i+1[1] [σ]) = j V \σ εj, σ j) d i x σ j ) d i+1 [1] [σ j] x σ j ) = εj, σ j) d i x 1,1 ) d i+1 x σ j ) d i+1 x 1,1 ) d i+2 [1] [σ j] x σ ) j V \σ = x d i+1,1 εj, σ j) d i x σ j ) x d i+2,1 d i+1 [1] [σ j] x σ ) j V \σ = x ) d i+1,1 id,i [1] [σ]). 21d) x d i+2,1

14 14 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN 8.2. Eigenvectors of cones. Our next goal is to describe the Laplacian eigenvalues and eigenvectors of 1 Λ in terms of those of Λ. Theorem 8.1. As before, let Λ be a d-dimensional simplicial complex on V = [2, n], and let = 1 Λ. Then i ) = X d i+1,1 + λ: λ i Λ), λ 0 } X d i+1,1 + X } d i+1,1 µ: µ X i 1Λ), µ 0 22) d i+2,1 X d i+1,1 } β iλ). Here the symbol denotes multiset union, and the superscript in the last line of 22) indicates multiplicity. Proof. Throughout the proof, we abbreviate L = L ud i ); all other Laplacians that arise will be specified precisely. First, let V Ci ud and, by Lemma 7.2, Λ) be an eigenvector of L ud Using 21a)... 21d) and 23), we calculate Λ,i with eigenvalue λ 0. Then Lud Λ,i V ) = λ V, Λ,i V ) = 1 λ Λ,i Λ,i+1 Λ,i+1 V )) ) = 0. 23) L [ ] V ) =,i,i[ ] V )) =,i d i Γ[ ] V + [ ] Λ,i+1 V ) ) = x d i+1,1,i+1 [1] V ) +,i+1 [ ] Λ,i+1 V ) ) = x d i+1,1 d i Γ [1] V x ) d i+1,1 [1] Λ,i V ) x d i+2,1 + [ ] Λ,i+1 Λ,i+1 V )) = X d i+1,1 + λ) [ ] V ). 24) Therefore, [ ] V C i ) is an eigenvector of L, with eigenvalue X d i+1,1 + λ. Notice that this eigenvalue cannot be zero: since 1 Λ, no Laplacian eigenvalue of Λ can possibly equal X d i,1. Second, let W be an eigenvector in Ci du Λ) with nonzero eigenvalue µ. That is, L du Λ i W ) = Λ,i Λ,i W )) = µw, and Λ,i+1W ) = 0 by a computation similar to 23). Define these are both nonzero elements of C i ). Then A = [ ] W, B = [1] Λ,i W ); LA) =,i+1,i+1 [ ] W ) ) =,i+1 d i Γ[ ] W + [ ] Λ,i+1 W ) ) = x d i+1,1,i+1 [1] W ) 25a) = x d i+1,1 d i Γ [1] W x ) d i+1,1 [1] Λ,i W ) x d i+2,1 ) Xd i+1,1 = X d i+1,1 A B 25b) x d i+2,1

15 SIMPLICIAL MATRIX-TREE THEOREMS 15 and LB) =,i+1,i+1 [1] Λ,i W )) ) =,i+1 x ) d i+1,1 [1] x Λ,i Λ,i W )) d i+2,1 = x d i+1,1 µ,i+1 [1] W ) x d i+2,1 = X ) ) d i+1,1 Xd i+1,1 µ A + µ B, x d i+2,1 X d i+2,1 where we have reused the equality between 25a) and 25b). Letting [uh oh!! Here it will be hard to use p for the first vertex!] p = X d i+1,1, the calculations above say that which implies that LA) = pa + qb, q = X d i+1,1 x d i+2,1, r = µ x d i+2,1, LB) = rpa + qb), LpA + qb) = pla) + qlb) = ppa + qb) + qrpa + qb) = p + qr)pa + qb). That is, pa + qb is an eigenvector of L with eigenvalue p + qr = X d i+1,1 + X d i+1,1 X d i+2,1 µ. As before, this quantity cannot be zero because Λ does not contain the vertex 1. Third, let Z C 0 i Λ); we will show that [ ] Z is an eigenvector of L with eigenvalue X d i+1,1. Indeed, by 2) of Proposition 7.3, we have Λ,i Z) = Λ,i+1Z) = 0, so that Therefore Λ,i Z) = Λ,i+1 Z) = 0. L[ ] Z) =,i+1,i+1[ ] Z)) =,i+1 d i Γ[ ] Z) + [ ] Λ,i+1 Z) ) = x d i+1,1,i+1 [1] Z) = x d i+1,1 d i Γ [1] Z x ) d i+1,1 [1] Λ,i Z) x d i+2 = X d i+1,1 [ ] Z), 26) as desired. By assertion 3) of Proposition 7.3, the multiplicity of this eigenvalue is dim C 0 i Λ) = β i Λ), At this point, we have proven that 22) is true if = is replaced with. On the other hand, we have accounted for dim C ud i Λ) + dim C du Λ) + dim Ci 0 Λ) = dim C i Λ) = f i Λ) i nonzero eigenvalues in i ). Since the preceding calculations hold for all i, we also know f i 1 Λ) nonzero eigenvalues of L ud,i 1 = L du,i ). By 20b), we have accounted for all f iλ) + f i 1 Λ) = f i ) = dim C i ) eigenvalues of L tot,i. So we have indeed found all the nonzero eigenvalues of L. One can obtain explicit formulas for the spectra s du i ) and s tot i ) by applying 20a) and 20b) to the formula 22); we omit the details.

16 16 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN 8.3. The vertex weighting and near-cones. In this section, we mimic the setup and proof of Lemma 5.3 of [9] to obtain a recurrence, Proposition 8.5, for the Laplacian eigenvalues of near-cones in terms of the eigenvalues of the deletion and link of the apex. Note that, by itself, Proposition 8.5 is not a proper recurrence, per se, in that it only computes the eigenvalues of a pure complex in terms of not necessarily pure complexes, and so it cannot be applied recursively. The proper recurrence for shifted complexes will have to wait until later! put in proper reference once structure of paper is more firmed up. Since we will be comparing complexes with similar face sets but of different dimensions, we begin by describing how their Laplacian spectra are related. We will need the lowering operator defined as the inverse of : that is, x i,j = x i 1,j, extended to a ring endomorphism on k. If S is a set of rational functions, we write S for f : f S}. [Question: Why not just write Lemma 8.2 as j ) = d i i))?] j Lemma 8.2. Let be a simplicial complex of dimension d, and let j < i d. Then j i) ) = d i ). Proof. The complexes i) and have the same face sets for every dimension i, but dim i) = dim d i). Therefore, the vertex-weighted boundary maps and Laplacians of i) can be obtained from those of by applying d i, from which the lemma follows. [Note: Maybe Lemma 8.3 and Corollary 8.4 should change to, not Σ. Also need a reminder of definition of pure skeleton.] Lemma 8.3. If Σ d is a d-dimensional simplicial complex, then d 1Σ) = d 1Σ [d] ). Proof. This result is proved in [8, Lemma 3.2], but we sketch the proof here for completeness. First note that L ud d 1 depends only on d 1)- and d-dimensional faces. In these two dimensions, Σ and Σ [d] differ only in some possible) additional d 1)-dimensional faces of Σ that are not contained in any d-dimensional faces of Σ. As a result, for each of these faces σ, we have L ud Σ,d 1σ = 0, and for all other d 1)-dimensional faces τ of both complexes, L ud Σ,d 1 τ = Lud Σ [d],d 1τ. The lemma now follows readily. Corollary 8.4. If Σ d is a d-dimensional simplicial complex, then Proof. By Lemmas 8.2 and 8.3, i 1Σ) = d i i 1Σ [i] ). i 1Σ) = d i i 1Σ i) ) = d i i 1Σ i) ) [i] ) = d i since Σ i) ) [i] = Σ [i]. j i 1Σ [i] ), Proposition 8.5. Let Σ d be a pure near-cone with apex p, and let = del p Σ and Λ = link p Σ be the deletion and link, respectively, of Σ with respect to vertex p. Then d 1Σ) = X 1,p + λ: λ d 1 ), λ 0 } X 1,p + X } 1,p µ: µ X d 2Λ), µ 0 2,p X 1,p } β d 1 ). Proof. If Σ is a cone, then the proposition follows from direct application of Theorem 8.1. Note that d 1 ) consists of only 0 s in this case, since is only d 1)-dimensional.) Thus, we may as well assume for the remainder of the proof that Σ is not a cone. In this case, dim = d and dim Λ = d 1. We now claim that d 1) = Λ. 27)

17 SIMPLICIAL MATRIX-TREE THEOREMS 17 To prove this claim, first note that Λ is obvious, and so Λ d 1), since dim Λ = d 1. Next, if F d 1), then p F, F Σ, and dim F d 1. By purity, F is in some d-dimensional face, so F v Σ for some v V. Since F v Σ and Σ is a near-cone, F p Σ, so F Λ, which establishes the claim. We next claim that Σ d) = Σ = p ) d). 28) To show that Σ p ) d), it suffices to show Σ p ), since Σ is only d-dimensional. To this end, let F Σ. If F, we are done, so assume F, which means p F. We may thus let F = F p for some F, but this immediately shows F p. To show that p ) d) Σ, let F p ) d). If F, we are done, since Σ, so assume F. Then F = F p for some F, and dim F d 1 since dim F d. By purity then, F v Σ for some v V. Since F v Σ and Σ is a near-cone, F = F p Σ. This establishes the second claim. Therefore, by Lemma 8.2 applied to equation 28) keeping in mind dim p = d + 1), and then by Theorem 8.1, d 1Σ) = d 1p ) = X 2,p + λ: λ d 1 ), λ 0 } X 2,p + X 2,p X 3,p µ: µ d 2 ), µ 0 X 2,p } β d 1 ) = X 1,p + λ: λ d 1 ), λ 0 } X 1,p + X } 1,p µ: µ X d 2 ), µ 0 2,p X 1,p } β d 1 ). } The proposition now follows from Lemma 8.2 again, this time applied to equation 27). [Note: This would be the place for Corollary 9.3 ] 9. z-polynomials and critical pairs 9.1. z-polynomials. Let S and T be multisets of integers. Define a Laurent polynomial zs, T ) by the formula zs, T ) = 1 X S j, 29) X S where as usual the symbol denotes multiset union. Proposition 9.1. Let d > i be integers, and let S, T be sets of integers greater than p. Then the following identities hold: and where S = S p}, T = T p}. j T X d i+1,p + d i zs, T ) = d i zs, T ) 30) X d i+1,p + X d i+1,p X d i+2,p d i+1 zs, T ) = d i z S, T ) 31)

18 18 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN Proof. We will use the identity 14a) [or some generalization thereof, to deal with more general indices] repeatedly in the calculations. First, and second, X d i+1,p + d i 1 zs, T ) = X d i+1,p + d i+1 d i X S j X S j T 1 = X d i+1,p d i+1 d i+1 X S + d i X S j X S j T 1 = d i d i+1 X X S + d i X S j S j T 1 = d i+1 d i X S j X S = d i zs, T ), j T X d i+1,p + X d i+1,p d i+1 X d i+1,p zs, T ) = X d i+1,p + X d i+2,p X d i+2,p d i+2 d i+1 X S j X S = X d i+1,p + X d i+1,p d i+1 d i+1 X S j X S j T 1 = X d i+1,p d i+1 d i+1 X X S + X d i+1,p d i+1 X S j S j T 1 = d i d i+1 X X S p + d i X S j S j T 1 = d i+1 d i X X S j S = d i z S, T ). j T Theorem 9.2. Let Σ d be a shifted simplicial complex. determined recursively as follows. Every nonzero eigenvalue of i 1 Σ) has the form of a z-polynomial. If Σ has no vertices, then i 1 Σ) consists only of 0 s. Otherwise, i 1Σ) = d i zs, T } ): 0 zs, T ) i 1 ) d i z S, T } ): 0 zs, T ) i 2Λ) j T The nonzero eigenvalues of i 1 Σ) are d i z, ) } 32) βi 1 ), where p is the initial vertex of Σ [i], S = S p, T = T p, = del p Σ [i], and Λ = link p Σ [i]. Proof. The proof is by induction on the number of vertices of Σ. When Σ has no vertices, Σ is either the empty simplicial complex with no faces, or the simplicial complex whose only face is the empty face. In either case, it is not hard to see that i 1 Σ) consists only of 0 s.

19 SIMPLICIAL MATRIX-TREE THEOREMS 19 So now we may assume that Σ has at least one vertex, and then i 1Σ [i] ) = X 1,p + λ: λ i 1 ), λ 0 } X 1,p + X } 1,p µ: µ X i 2Λ), µ 0 2,p X 1,p + 0} β i 1 ) = X 1,p + zs, T ): zs, T ) i 1 ), zs, T ) 0 } X 1,p + X } 1,p zs, T ): zs, T ) X i 2Λ), zs, T ) 0 2,p X 1,p + z, )} β i 1 ) = zs, T } ): 0 zs, T ) i 1 ) z S, T } ): 0 zs, T ) i 2Λ) )} βi 1 ) z,. The =-equivalence above is by Proposition 8.5. The equality following it uses the identity z, ) = 0 and induction, since and Λ each have one fewer vertex than Σ. Note that λ and µ are each replaced by zs, T ), with no operator, because dim i and dim Λ = i 1. The final equality is by Proposition 9.1. The result now follows from Corollary Critical pairs. Definition 9.3. A critical pair for a k-family F is an ordered pair A, B), where A = a 1 < a 2 < < a k } and B = b 1 < b 2 < < b k } are sets of integers such that A F, B F, and B covers A in componentwise order. That is, b i = a i + 1 for exactly one i, and b j = a j for all j i. Note that b i might not be in the vertex set of F. The signature σa, B) of A, B) is the set of vertices a 1,..., a i 1, a i }. The long signature τ p A, B) is the ordered pair of sets S, T ) where S = a 1,..., a i 1 } and T = j : p j a i }. The set of critical pairs is especially significant for a shifted family and by extension, for a shifted complex). Since a shifted family is an order ideal with respect to <, members of the family lie below the critical pairs, non-members lie above, and the critical pairs themselves identify the exact boundary between the two in the Hasse diagram of <. [Note: Possibly a reference to Carly s Obstructions to shiftedness paper may be nice here, especially if we can tie in to critical pairs where 0 is replaced by 1.] [Note: Maybe use v instead of j for typical vertex below.] Definition 9.4. The degree of vertex j in family F is deg F j) = F F : j F }. Proposition 9.5. If F is a shifted family, then deg F j) deg F j+1) counts the number of signatures of F whose greatest element is j. This proposition is used in the recursion below, and will also later! show how the results in this section specialize to the more familiar formula for unweighted eigenvalues in the obvious way.

20 20 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN Proof. Let S = F F : j F } and T = F F : j + 1 F }. Partition S = S 1 S 2 S 3 and T = T 1 T 2 T 3 as follows: S 1 = F F : j, j + 1 F } S 2 = F F : j F, j + 1 F, F j j + 1) F} S 3 = F F : j F, j + 1 F, F j j + 1) F} T 1 = F F : j, j + 1 F } T 2 = F F : j F, j + 1 F, F j + 1) j F} T 3 = F F : j F, j + 1 F, F j + 1) j F}. Then S 1 = T 1, and there is an obvious bijection between S 2 and T 2. Since F is shifted, T 3 =, so deg F j) deg F j + 1) = S T = S 3. Now note that if F S 3, then F, G) is a critical pair, where G = F j j + 1). In each such critical pair, the greatest element of the signature is j. Conversely, if A, B) is a critical pair whose signature s greatest element is j, then A S 3. Later, we ll show the specialization hinted at above before the proof). For now, a quick corollary we ll need shortly. Corollary 9.6. If Σ is a pure d-dimensional shifted simplicial complex with initial vertex p, then deg Σd p) deg Σd p + 1) = β d 1 del p Σ). Proof. As in the proof of Proposition 9.5 above, and, in this case, deg Σd p) deg Σd p + 1) = S 3, S 3 = F Σ d : p F, p + 1 F, F p p + 1) Σ d }. There is an obvious bijection between S 3 and S 3 = G del p Σ) d 1 : p + 1 G, G p + 1) del p Σ} given by G = F p. Then, by equation 10), β d 1 del p Σ) = S 3. Combining the four displayed equations yields the corollary. [Note: We need to define σ acting on a whole family, in the obvious way.] Proposition 9.7. Let Σ be a shifted complex with initial vertex p. Let = del p Σ and Λ = link p Σ. Then σσ i ) = σ i ) p F : F Λ i 1 } p} deg Σ i p) deg Σi p+1), where denotes multiset union. Proof. First note that Proposition 9.5 immediately implies that the signature p} occurs with the desired multiplicity. If A i, B i, and B covers A in the componentwise partial order, then A Σ i and p A. We will show B Σ i. But since p A and A < B, we conclude p B, and so B i implies B Σ i. This establishes the multiset injection σ i ) σσ i ). If A Λ i 1, B Λ i 1, and B covers A in the componentwise partial order, then p A and p A Σ i. We will show that p B Σ i ; thus the critical pair A, B) in Λ i 1 gives rise to the critical pair p A, p B) in Σ i, and it is clear σp A, p B) = p σa, B). Since A < B, then p A < p B. Furthermore, B Λ i 1 implies p B Σ i. This establishes the multiset injection p F : F σλ i 1 )} σσ i ). Now we must show, conversely, that every signature F p} of Σ i arises from σ i ) or p F : F σλ i 1 )} as described above. Let F p} be a signature of Σ i, with critical pair

21 SIMPLICIAL MATRIX-TREE THEOREMS 21 A, B), so A Σ i, B Σ i, and B covers A in the componentwise partial order. If p F, then p A and so A i. Also B i, since B Σ i. Thus A, B) is a critical pair for i. Otherwise, p F, so F = F p where F. In this case, we claim that p A and p B for the critical pair A, B) whose signature is F : p F directly implies p A; and then p B, since p B and p A would lead to the contradiction F = σa, B) = p}. Thanks to this claim, we may let A = p A and B = p B. From A Σ i it follows A Λ i 1, and similarly B Σ i implies B Λ i 1. Thus A, B) is a critical pair for Λ i 1. If family F has initial vertex p, then denote by ψf) the multiset of long signatures, with respect to p, of critical pairs of F. In other words, ψf) = τ p A, B): A, B) is a critical pair of F}. Corollary 9.8. Let Σ be a shifted simplicial complex. The multiset of long signatures τ p of its i-dimensional critical pairs are determined recursively as follows. If Σ has no vertices, then ψσ i ) =. Otherwise, ψσ i ) = S, T ): S, T ) ψ i )} S, T ): S, T ) ψλ i 1 )}, )} β i 1 ), where p is the initial vertex of Σ [i], S = S p, T = T p, = del p Σ [i], and Λ = link p Σ [i]. Proof. For the third set of the right-hand side, note that if σa, B) = p}, then τ p A, B) =, p}) =, ), and apply Corollary 9.6 to the pure i-dimensional shifted complex Σ [i]. For the first two sets, the only hard part is to note that if the first vertex of Σ i is p, then p + 1 is the first vertex of i and Λ i 1 unless i =, but in that case we don t have to worry about the first set). This accounts for the T s. Even though Proposition 9.7 above defines and Λ somewhat differently than here, it is not a problem because it is almost obvious that del p Σ) i = del p Σ [i] ) i and link p Σ) i 1 = link p Σ [i] ) i 1. Then, since and Λ only appear as i and Λ i 1, it doesn t matter whether we set them to be the deletion and link of Σ or of Σ [i]. [Note: This proof feels a little sketchy, but I m not sure what else to say, or how to say it. At least not yet.] [Somewhere: note that Σ [i] is shifted when Σ is, or even just when Σ i is.] 9.3. The theorem, and its consequences. Theorem 9.9. Let Σ d be a shifted simplicial complex. Then, for each 0 i d, i 1Σ) = d i zs, T ): S, T ) ψσ i )}. Proof. Here see ye two recurrences, and lo! they are the same. Theorem 9.2 and Corollary 9.8.) Alternate proof. The proof is by induction on the number of vertices of Σ. When Σ has no vertices, Σ is either the empty simplicial complex with no faces, or the simplicial complex whose only face is the empty face. In either case, it is not hard to see that both sides of the =-equivalence are empty.

22 22 ART M. DUVAL, CAROLINE KLIVANS, AND JEREMY L. MARTIN So now we may assume that Σ has at least one vertex. We will first work with the pure i-skeleton Σ [i]. Assume the initial vertex of Σ [i] is p. Let = del p Σ [i] and Λ = link p Σ [i]. Then i 1Σ [i] ) = X 1,p + λ: λ i 1 ), λ 0 } X 1,p + X } 1,p µ: µ X i 2Λ), µ 0 2,p X 1,p + 0} β i 1 ) = X 1,p + zs, T ): S, T ) ψ i )} X 1,p + X } 1,p zs, T ): S, T ) ψλ i 1 ) X 2,p X 1,p + z, )} β i 1 ) = zs, T } ): S, T ) ψ i ) z S, T } ): S, T ) ψλ i 1 ) z, ) } βi 1 ) = zs, T ): S, T ) ψσ [i] ) i ) }. The =-equivalence above is by Proposition 8.5. The first equality after the =-equivalence) uses the identity z, ) = 0 and induction, since and Λ each have one fewer vertex than Σ, and since dim = i and dim Λ = i 1. More precisely, dim i, but if dim < i, then i 1 ) is empty.) The next equality is by Proposition 9.1. The final equality is by Proposition 9.8 with Corollary 9.6. We now conclude, by Corollary 8.4, that i 1Σ) = d i i 1Σ [i] ) = d i zs, T ): S, T ) ψσ [i] ) i ) } = d i zs, T ): S, T ) ψσ i )}. Here we show how Theorem 9.9 specializes to the previously known formula from [9], and a generalization of that formula to coarse weighting. Definition If λ is a partition, let X λ be the multiset where each part i of λ is replaced by X i := X X i. Remark It is standard partition theory isn t it?) that λ, the conjugate of partition λ may be described as the partition whose part t occurs with multiplicity λ t λ t+1. Corollary In the coarse weighting, ŝ ud d 1Σ) = X degσd ), where Σ is a shifted d-dimensional complex with initial vertex 1, and deg Σd denotes the partition of its degree sequence Proof. To reduce to coarse weighting, set X i,j = X j, so zs, T ) = j T X j = X X t = X t where T = [1, t]. In this case, if A, B) is a critical pair of Σ d, and if τ 1 A, B) = S, T ), then zs, T ) = X t, where t = max σa, B). By Theorem 9.9, then, d 1 Σ) is =-equivalent to the partition where X t occurs with multiplicity A, B): A, B) is a critical pair of Σ d, t = max σa, B)}. By Proposition 9.5 that multiplicity may be counted by deg Σd t) deg Σd t + 1). The corollary now follows from Remark 9.11

23 SIMPLICIAL MATRIX-TREE THEOREMS 23 The formula in [9] follows immediately from this corollary An example: the bipyramid. To illustrate Theorem 8.1 and Proposition 8.5, [just Proposition 8.5, I think] we calculate the top-dimensional up-down Laplacian spectrum for the bipyramid complex B = 123, 124, 125, 134, 135, 234, 235. The bipyramid is shifted, which is equivalent to the condition that it is an iterated near-cone. [Citation?] In our recursive calculation, we will encounter the following subcomplexes of B: B 1 = B, B 5 = 3, 4, 5, B 2 = 234, 235, B 6 = 4, 5, B 3 = 23, 24, 25, 34, 35, B 7 = 5, B 4 = 34, 35, B 8 = } B 1 B 2 B 3 B 4 B 5 B 6 [Notes for diagram: Can we add B 7? Can we make 3 the middle vertex in B 4 and B 5? In B 4, this is necessary, so then it would be nice to change B 5 for consistency.) In B 1, the center triangle needs to be vertices 1,2,3. For consistency, I would recommend putting 4 at the top, 5 on the bottom, 2 on the left, 1 in the lower) right and 3 in the upper) right.] Observe that B 1 is a near-cone with apex 1, del 1 B 1 = B 2, and link 1 B 1 = B 3 ; B 2 = 2 B 4 ; B 3 is a near-cone with apex 2, del 2 B 3 = B 4, and link 2 B 3 = B 5 ; B 4 = 3 B 6 ; B 5 is a near-cone with apex 3, del 3 B 5 = B 6, and link 3 B 5 = B 8 ; B 6 is a near-cone with apex 4, del 4 B 6 = B 7, and link 4 B 6 = B 8 ; and B 7 = 4 B 8. [Note: We might get rid of the calculation of eigenvalues without z s.] First, B 8 has no vertices, and has only a 0 eigenvalue. Second, to calculate 1B 7 ), we apply Theorem 8.1 with Σ = B 7, Λ = B 8, and d = i = 1 [or Proposition 8.5 with Σ = B 7, = Λ = B 8 and d = 0]. Note that 2Λ) = 1Λ) = and β 1 Λ) = 1, [for Proposition 8.5, note that 2Λ) = 1 ) = and β 1 ) = 1] so we obtain 1B 7 ) = X 15 } 1. 33) Third, to calculate 1B 6 ), we apply Proposition 8.5 with Σ = B 6, = B 7, Λ = B 8, and d = 0. Note that 2Λ) = and β 1 ) = 0, so we obtain 1B 6 ) = X 14 + λ: λ 1B 7 ), λ 0 } = X 14 + X 15 }. 34)