Art Duval 1 Caroline Klivans 2 Jeremy Martin 3 1 University of Texas at El Paso 2 University of Chicago 3 University of Kansas AMS Central Section Meeting Special Session on Geometric Combinatorics DePaul University October 5, 2007
simplicial spanning trees Definition of simplicial spanning trees Let be a d-dimensional simplicial complex. Υ is a simplicial spanning tree of when: 0. Υ (d 1) = (d 1) ( spanning ); 1. H d (Υ; Z) = 0 ( acyclic ); 2. H d 1 (Υ; Q) = 0 ( connected ); 3. f d (Υ) = f d ( ) β d ( ) + β d 1 ( ) ( count ). If 0. holds, then any two of 1., 2., 3. together imply the third condition. When d = 1, coincides with usual definition.
simplicial spanning trees Metaconnectedness Denote by T ( ) the set of simplicial spanning trees of. Proposition T ( ) iff is metaconnected (homology type of wedge of spheres). Many interesting complexes are metaconnected, including everything we ll talk about.
Theorems Simplicial Matrix-Tree Theorem Version II d = metaconnected simplicial complex Γ T ( (d 1) ) Γ = restriction of d to faces not in Γ reduced Laplacian L Γ = Γ Γ Theorem [DKM, 2006] h d = Υ T ( ) H d 1 (Υ) 2 = H d 2 ( ; Z) 2 H d 2 (Γ; Z) 2 det L Γ.
Theorems Weighted Simplicial Matrix-Tree Theorem Version II d = metaconnected simplicial complex Introduce an indeterminate x F for each face F Weighted boundary : multiply column F of (usual) Γ T ( (d 1) ) Γ = restriction of d to faces not in Γ by x F Weighted reduced Laplacian L = Γ Γ Theorem [DKM, 2006] h d = Υ T ( ) H d 1 (Υ) 2 F Υ x 2 F = H d 2 ( ; Z) 2 H d 2 (Γ; Z) 2 det L Γ.
Definition Definition of shifted complexes Vertices 1,..., n F, i F, j F, i < j F i j Equivalently, the k-faces form an initial ideal in the componentwise partial order. Example (bipyramid with equator) 123, 124, 125, 134, 135, 234, 235
Definition Hasse diagram 146 236 245 136 145 235 126 135 234 125 134 124 123
Definition Hasse diagram 146 236 245 136 145 235 126 135 234 125 134 124 123
Definition Links and deletions Deletion, del 1 = {G : 1 G, G }. Link, lk 1 = {F 1: 1 F, F }. Deletion and link are each shifted, with vertices 2,..., n. Example: = 123, 124, 125, 134, 135, 234, 235 del 1 = 234, 235 lk 1 = 23, 24, 25, 34, 35
Fine weightings The Combinatorial fine weighting Let d be a shifted complex on vertices [n]. For each facet A = {a 1 < a 2 < < a d+1 }, define x A = d+1 i=1 x i,ai. Example If Υ = 123, 124, 134, 135, 235 is a simplicial spanning tree of, its contribution to h 2 is (x 1,1 x 2,2 x 3,3 )(x 1,1 x 2,2 x 3,4 )(x 1,1 x 2,3 x 3,4 )(x 1,1 x 2,3 x 3,5 )(x 1,2 x 2,3 x 3,5 )
Fine weightings From Combinatorial to Algebraic In Weighted Simplicial Matrix Theorem II, pick Γ to be the set of all (d 1)-dimensional faces containing vertex 1.
Fine weightings From Combinatorial to Algebraic In Weighted Simplicial Matrix Theorem II, pick Γ to be the set of all (d 1)-dimensional faces containing vertex 1. H d 2 (Γ; Z) and H d 2 ( ; Z) are trivial, so, h d = det L Γ
Fine weightings From Combinatorial to Algebraic In Weighted Simplicial Matrix Theorem II, pick Γ to be the set of all (d 1)-dimensional faces containing vertex 1. H d 2 (Γ; Z) and H d 2 ( ; Z) are trivial, so, by some easy linear algebra, h d = det L Γ = ( X σ ) det(x 1,1 I + ˆL del1,d 1) σ lk 1 where ˆL is an algebraic fined weighted Laplacian.
Fine weightings From Combinatorial to Algebraic In Weighted Simplicial Matrix Theorem II, pick Γ to be the set of all (d 1)-dimensional faces containing vertex 1. H d 2 (Γ; Z) and H d 2 ( ; Z) are trivial, so, by some easy linear algebra, h d = det L Γ = ( X σ ) det(x 1,1 I + ˆL del1,d 1) σ lk 1 = ( σ lk 1 X σ )( λ e val of ˆL del1,d 1 X 1,1 + λ), where ˆL is an algebraic fined weighted Laplacian.
Fine weightings The Algebraic fine weighted boundary map For faces A B with dim A = i 1, dim B = i, define X AB = d i x B d i+1 x A where x i,j = x i+1, j. Construct weighted boundary map by multiplying (A, B) entry of usual boundary map by X AB. Example: X (235,25) = x 12x 23 x 35 x 22 x 35 Weighted boundary maps satisfy = 0.
Definitions Critical pairs Definition A critical pair of a shifted complex d is an ordered pair (A, B) of (d + 1)-sets of integers, where A and B ; and B covers A in componentwise order.
Definitions Critical pairs 146 236 245 136 145 235 126 135 234 125 134 124 123
Definitions Critical pairs 146 236 245 136 145 235 126 135 234 125 134 124 123
Definitions Critical pairs 146 236 245 136 145 235 126 135 234 125 134 124 123
Definitions The Signature of a critical pair Let (A, B) be a critical pair of a complex : Definition Example A = {a 1 < a 2 < < a i < < a d+1 }, B = A \ {a i } {a i + 1}. The signature of (A, B) is the ordered pair ( {a1, a 2,..., a i 1 }, a i ). = 123, 124, 125, 134, 135, 234, 235 (the bipyramid) critical pair (125,126) (135,136) (135,145) (235,236) (235,245) signature
Definitions The Signature of a critical pair Let (A, B) be a critical pair of a complex : Definition Example A = {a 1 < a 2 < < a i < < a d+1 }, B = A \ {a i } {a i + 1}. The signature of (A, B) is the ordered pair ( {a1, a 2,..., a i 1 }, a i ). = 123, 124, 125, 134, 135, 234, 235 (the bipyramid) critical pair signature (125,126) (12,5) (135,136) (135,145) (235,236) (235,245)
Definitions The Signature of a critical pair Let (A, B) be a critical pair of a complex : Definition Example A = {a 1 < a 2 < < a i < < a d+1 }, B = A \ {a i } {a i + 1}. The signature of (A, B) is the ordered pair ( {a1, a 2,..., a i 1 }, a i ). = 123, 124, 125, 134, 135, 234, 235 (the bipyramid) critical pair signature (125,126) (12,5) (135,136) (13,5) (135,145) (235,236) (235,245)
Definitions The Signature of a critical pair Let (A, B) be a critical pair of a complex : Definition Example A = {a 1 < a 2 < < a i < < a d+1 }, B = A \ {a i } {a i + 1}. The signature of (A, B) is the ordered pair ( {a1, a 2,..., a i 1 }, a i ). = 123, 124, 125, 134, 135, 234, 235 (the bipyramid) critical pair signature (125,126) (12,5) (135,136) (13,5) (135,145) (1,3) (235,236) (235,245)
Definitions The Signature of a critical pair Let (A, B) be a critical pair of a complex : Definition Example A = {a 1 < a 2 < < a i < < a d+1 }, B = A \ {a i } {a i + 1}. The signature of (A, B) is the ordered pair ( {a1, a 2,..., a i 1 }, a i ). = 123, 124, 125, 134, 135, 234, 235 (the bipyramid) critical pair signature (125,126) (12,5) (135,136) (13,5) (135,145) (1,3) (235,236) (23,5) (235,245)
Definitions The Signature of a critical pair Let (A, B) be a critical pair of a complex : Definition Example A = {a 1 < a 2 < < a i < < a d+1 }, B = A \ {a i } {a i + 1}. The signature of (A, B) is the ordered pair ( {a1, a 2,..., a i 1 }, a i ). = 123, 124, 125, 134, 135, 234, 235 (the bipyramid) critical pair signature (125,126) (12,5) (135,136) (13,5) (135,145) (1,3) (235,236) (23,5) (235,245) (2,3)
Eigenvalues Finely Weighted Laplacian Eigenvalues Theorem [DKM 2007] Let be a shifted complex. Then the finely weighted Laplacian eigenvalues of are specified completely by the signatures of critical pairs of. signature (S, a) eigenvalue 1 X S a j=1 X S j
Eigenvalues Examples of finely weighted eigenvalues Critical pair (135,145); signature (1,3): X 11 X 21 + X 11 X 22 + X 11 X 23 X 21
Eigenvalues Examples of finely weighted eigenvalues Critical pair (135,145); signature (1,3): X 11 X 21 + X 11 X 22 + X 11 X 23 X 21 Critical pair (235,236); signature (23,5): X 11 X 22 X 33 + X 12 X 22 X 33 + X 12 X 23 X 33 + X 12 X 23 X 34 + X 12 X 23 X 35 X 22 X 33
Eigenvalues Corollaries Generalizes D.-Reiner formula for eigenvalues of shifted complexes in terms of degree sequences. (The a of the signatures are the entries of the conjugate degree sequence.)
Eigenvalues Corollaries Generalizes D.-Reiner formula for eigenvalues of shifted complexes in terms of degree sequences. (The a of the signatures are the entries of the conjugate degree sequence.) We can reconstruct a shifted complex from its finely weighted eigenvalues, so we can hear the shape of a shifted complex, at least if our ears are fine enough.
Enumeration Deletion-link recursion We can compute signatures recursively, from deletion and link, as follows: Each (S, a) from del 1 is also a signature of. Each (S, a) from lk 1 becomes signature (S 1, a) of. Additionally, β d 1 (del 1 ) copies of (, 1). Example complex = 123, 124, 125, 234, 235 signature
Enumeration Deletion-link recursion We can compute signatures recursively, from deletion and link, as follows: Each (S, a) from del 1 is also a signature of. Each (S, a) from lk 1 becomes signature (S 1, a) of. Additionally, β d 1 (del 1 ) copies of (, 1). Example complex = 123, 124, 125, 234, 235 del 1 = 234, 235 lk 1 = 23, 24, 25, 34, 35 signature
Enumeration Deletion-link recursion We can compute signatures recursively, from deletion and link, as follows: Each (S, a) from del 1 is also a signature of. Each (S, a) from lk 1 becomes signature (S 1, a) of. Additionally, β d 1 (del 1 ) copies of (, 1). Example complex signature = 123, 124, 125, 234, 235 del 1 = 234, 235 {(2, 3), (23, 5)} lk 1 = 23, 24, 25, 34, 35 {(2, 5), (3, 5), (, 3)}
Enumeration Deletion-link recursion We can compute signatures recursively, from deletion and link, as follows: Each (S, a) from del 1 is also a signature of. Each (S, a) from lk 1 becomes signature (S 1, a) of. Additionally, β d 1 (del 1 ) copies of (, 1). Example complex signature = 123, 124, 125, 234, 235 {(2, 3), (23, 5), (12, 5), (13, 5), (1, 3)} del 1 = 234, 235 {(2, 3), (23, 5)} lk 1 = 23, 24, 25, 34, 35 {(2, 5), (3, 5), (, 3)}
Enumeration Finely weighted enumeration of SST s in shifted complexes Theorem h d = ( σ lk 1 X ) ( P a ) j=1 X S j σ 1 (S,a) sign.(del 1 ) X S 1. Example lk 1 = 23, 24, 25, 34, 35 del 1 = 234, 235 sign.(del 1 ) = {(2, 3), (23, 5)} h d ( ) = (X 123 X 124 X 134 X 125 X 135 ) ( ) ( ) X12 + X 22 + X 23 X123 + X 223 + X 233 + X 234 + X 235. X 12 X 123
Enumeration Corollary By specializing to d = 1, we get a formula from Martin-Reiner (itself a special case of a result due to Remmel and Williamson) of finely weighted enumeration of spanning trees of threshold graphs (1-dimensional shifted complexes).
Color-shifted complexes Definition of color-shifted complexes Set of colors n c vertices, (c, 1), (c, 2),... (c, n c ) of color c. Faces contain at most one vertex of each color. Can replace (c, j) by (c, i) in a face if i < j. Example: Faces written as (red,blue,green): 111, 112, 113, 121, 122, 123, 131, 132, 211, 212, 213, 221, 222, 223, 231,232.
Color-shifted complexes Conjecture for complete color-shifted complexes Let be the color-shifted complex generated by the face with red a, blue b, green c. Let the red vertices be x 1,..., x a, the blue vertices be y 1,..., y b, and the green vertices be z 1,..., z c. Conjecture a b c h d ( ) = ( x i ) b+c 1 ( y j ) a+c 1 ( z k ) a+b 1 i=1 i=1 j=1 j=1 k=1 a b c ( x i ) (b 1)(c 1) ( y j ) (a 1)(c 1) ( z k ) (a 1)(b 1) k=1
Color-shifted complexes Notes on conjecture This is with coarse weighting. Every vertex v has weight x v, and every face F has weight x F = v F x v. The case with two colors is a (complete) Ferrers graph, studied by Ehrenborg and van Willigenburg.
Matroid complexes Conjecture for Matroid Complexes h d again seems to factor nicely, though we can t describe it yet. Once again, with coarse weighting