A Simplicial matrix-tree theorem, II. Examples

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1 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

2 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.

3 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.

4 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.

5 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.

6 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

7 Definition Hasse diagram

8 Definition Hasse diagram

9 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

10 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 )

11 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.

12 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

13 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.

14 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.

15 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.

16 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.

17 Definitions Critical pairs

18 Definitions Critical pairs

19 Definitions Critical pairs

20 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

21 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)

22 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)

23 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)

24 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)

25 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)

26 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

27 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

28 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

29 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.)

30 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.

31 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

32 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

33 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)}

34 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)}

35 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 X X X 235. X 12 X 123

36 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).

37 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.

38 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

39 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.

40 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