Combinatorial interpretations for -vectors T. Kyle Petersen DePaul - - PowerPoint PPT Presentation

Combinatorial interpretations for γ-vectors

• T. Kyle Petersen

DePaul University

joint with Eran Nevo (Cornell), arXiv:0909.0694 San Francisco, January, 2010

Combinatorial interpretations for γ-vectors

Gal’s conjecture An example The Γ complex A conjecture

The f - and h-vectors

Let ∆ be an (n − 1)-dimensional simplicial complex, fk(∆) = number of faces of dimension k − 1 f (∆; t) :=

n

• k=0

fk(∆)tk (f0, f1, . . . , fn) is the f -vector h(∆; t) := (1 − t)nf (∆; t/(1 − t)) =

n

• k=0

hk(∆)tk (h0, h1, . . . , hn) is the h-vector

The f - and h-vectors

Let ∆ be an (n − 1)-dimensional simplicial complex, fk(∆) = number of faces of dimension k − 1 f (∆; t) :=

n

• k=0

fk(∆)tk (f0, f1, . . . , fn) is the f -vector h(∆; t) := (1 − t)nf (∆; t/(1 − t)) =

n

• k=0

hk(∆)tk (h0, h1, . . . , hn) is the h-vector f -vectors are characterized by the Kruskal-Katona inequalities

The f - and h-vectors

∆ :

The f - and h-vectors

∆ :

• ◮ f0 = 1

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6 ◮ f2 = 6

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6 ◮ f2 = 6

f (∆; t) = 1 + 6t + 6t2

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6 ◮ f2 = 6

f (∆; t) = 1 + 6t + 6t2 = (1 + 2t + t2)

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6 ◮ f2 = 6

f (∆; t) = 1 + 6t + 6t2 = (1 + 2t + t2) + 4t(1 + t)

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6 ◮ f2 = 6

f (∆; t) = 1 + 6t + 6t2 = (1 + 2t + t2) + 4t(1 + t) + t2(1)

The f - and h-vectors

∆ :

• ◮ f0 = 1

◮ f1 = 6 ◮ f2 = 6

f (∆; t) = 1 + 6t + 6t2 = (1 + 2t + t2) + 4t(1 + t) + t2(1) h(∆; t) = 1 + 4t + t2

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i,

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i, e.g., 1 + 3t + 7t2 + 3t3 + t4

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i, e.g., 1 + 3t + 7t2 + 3t3 + t4 1 + 4t + 6t2 + 4t3 + t4

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i, e.g., 1 + 3t + 7t2 + 3t3 + t4 1 + 4t + 6t2 + 4t3 + t4 −t(1 + 2t + t2)

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i, e.g., 1 + 3t + 7t2 + 3t3 + t4 1 + 4t + 6t2 + 4t3 + t4 −t(1 + 2t + t2) +3t2

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i, e.g., 1 + 3t + 7t2 + 3t3 + t4 1 + 4t + 6t2 + 4t3 + t4 −t(1 + 2t + t2) +3t2 1 + 3t + 7t2 + 3t3 + t4 = (1 + t)4 − t(1 + t)2 + 3t2

The γ-vector

If h(t) = n

i=0 hiti is symmetric, then there exist γi such that

h(t) =

n/2

• i=0

γiti(1 + t)n−2i, e.g., 1 + 3t + 7t2 + 3t3 + t4 1 + 4t + 6t2 + 4t3 + t4 −t(1 + 2t + t2) +3t2 1 + 3t + 7t2 + 3t3 + t4 = (1 + t)4 − t(1 + t)2 + 3t2 the vector (γ0, γ1, . . .) is called the γ-vector

Gal’s conjecture

For ∆ a sphere, Dehn-Sommerville relations say h(∆) is symmetric, hi = hn−i, and hence ∆ has a well-defined γ-vector, denoted γ(∆)

Gal’s conjecture

For ∆ a sphere, Dehn-Sommerville relations say h(∆) is symmetric, hi = hn−i, and hence ∆ has a well-defined γ-vector, denoted γ(∆)

Conjecture (Gal (2005))

If ∆ is a flag homology sphere, then γ(∆) is nonnegative

Gal’s conjecture

For ∆ a sphere, Dehn-Sommerville relations say h(∆) is symmetric, hi = hn−i, and hence ∆ has a well-defined γ-vector, denoted γ(∆)

Conjecture (Gal (2005))

If ∆ is a flag homology sphere, then γ(∆) is nonnegative

◮ implies the Charney-Davis conjecture

Gal’s conjecture

For ∆ a sphere, Dehn-Sommerville relations say h(∆) is symmetric, hi = hn−i, and hence ∆ has a well-defined γ-vector, denoted γ(∆)

Conjecture (Gal (2005))

If ∆ is a flag homology sphere, then γ(∆) is nonnegative

◮ implies the Charney-Davis conjecture ◮ true in dimension ≤ 4 and other interesting cases (e.g.,

barycentric subdivisions, Coxeter complexes)

Gal’s conjecture

For ∆ a sphere, Dehn-Sommerville relations say h(∆) is symmetric, hi = hn−i, and hence ∆ has a well-defined γ-vector, denoted γ(∆)

Conjecture (Gal (2005))

If ∆ is a flag homology sphere, then γ(∆) is nonnegative

◮ implies the Charney-Davis conjecture ◮ true in dimension ≤ 4 and other interesting cases (e.g.,

barycentric subdivisions, Coxeter complexes)

What do the entries of the γ-vector count?

Combinatorial interpretations for γ-vectors

Gal’s conjecture An example The Γ complex A conjecture

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t)

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 132 213 231 312 321

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 213 231 312 321

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 1 t 213 231 312 321

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 1 t 213 1 t 231 312 321

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 1 t 213 1 t 231 1 t 312 321

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 1 t 213 1 t 231 1 t 312 1 t 321

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 1 t 213 1 t 231 1 t 312 1 t 321 2 t2

Eulerian polynomials

Let w ∈ Sn+1 and d(w) := |{i : wi > wi+1}|. Then, An(t) =

• w∈Sn+1

td(w) = h(∆(An); t) w d(w) td(w) 123 1 132 1 t 213 1 t 231 1 t 312 1 t 321 2 t2 A2(t) = 1 + 4t + t2 = h(∆(A2); t)

Eulerian polynomials

We have: A1(t) = 1 + t A2(t) = 1 + 4t + t2 A3(t) = 1 + 11t + 11t2 + t3 A4(t) = 1 + 26t + 66t2 + 26t3 + t4 . . .

Eulerian polynomials

We have: A1(t) = 1 + t = (1 + t) A2(t) = 1 + 4t + t2 A3(t) = 1 + 11t + 11t2 + t3 A4(t) = 1 + 26t + 66t2 + 26t3 + t4 . . .

Eulerian polynomials

We have: A1(t) = 1 + t = (1 + t) A2(t) = 1 + 4t + t2 = (1 + t)2 + 2t A3(t) = 1 + 11t + 11t2 + t3 A4(t) = 1 + 26t + 66t2 + 26t3 + t4 . . .

Eulerian polynomials

We have: A1(t) = 1 + t = (1 + t) A2(t) = 1 + 4t + t2 = (1 + t)2 + 2t A3(t) = 1 + 11t + 11t2 + t3 = (1 + t)3 + 8t(1 + t) A4(t) = 1 + 26t + 66t2 + 26t3 + t4 . . .

Eulerian polynomials

We have: A1(t) = 1 + t = (1 + t) A2(t) = 1 + 4t + t2 = (1 + t)2 + 2t A3(t) = 1 + 11t + 11t2 + t3 = (1 + t)3 + 8t(1 + t) A4(t) = 1 + 26t + 66t2 + 26t3 + t4 = (1 + t)4 + 22t(1 + t)2 + 16t2 . . .

The γ-vector for An

Define

• Sn = {w ∈ Sn : wn−1 < wn, and if wi−1 > wi then wi < wi−1}

The γ-vector for An

Define

• Sn = {w ∈ Sn : wn−1 < wn, and if wi−1 > wi then wi < wi−1}

Theorem (Foata-Sch¨ utzenberger (1970))

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w),

The γ-vector for An

Define

• Sn = {w ∈ Sn : wn−1 < wn, and if wi−1 > wi then wi < wi−1}

Theorem (Foata-Sch¨ utzenberger (1970))

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w), i.e., γi(An) = |{w ∈ Sn+1 : d(w) = i}|

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w)

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 1324 1423 2134 2314 2413 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1423 2134 2314 2413 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 2134 2314 2413 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 2314 2413 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 2413 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 1 t(1 + t) 2413 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 1 t(1 + t) 2413 1 t(1 + t) 3124 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 1 t(1 + t) 2413 1 t(1 + t) 3124 1 t(1 + t) 3412 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 1 t(1 + t) 2413 1 t(1 + t) 3124 1 t(1 + t) 3412 1 t(1 + t) 4123

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 1 t(1 + t) 2413 1 t(1 + t) 3124 1 t(1 + t) 3412 1 t(1 + t) 4123 1 t(1 + t)

The γ-vector for An

An(t) =

• w∈b

Sn+1

td(w)(1 + t)n−2d(w) w d(w) td(w)(1 + t)n−2d(w) 1234 (1 + t)3 1324 1 t(1 + t) 1423 1 t(1 + t) 2134 1 t(1 + t) 2314 1 t(1 + t) 2413 1 t(1 + t) 3124 1 t(1 + t) 3412 1 t(1 + t) 4123 1 t(1 + t) A3(t) = (1 + t)3 + 8t(1 + t) = 1 + 11t + 11t2 + t3

Kruskal-Katona inequalities

Observe that γ(A1) = (1) γ(A2) = (1, 2) γ(A3) = (1, 8) γ(A4) = (1, 22, 16) . . . are all Kruskal-Katona vectors

Combinatorial interpretations for γ-vectors

Gal’s conjecture An example The Γ complex A conjecture

The complex Γ(An)

We can identify elements of Sn+1 with faces a simplicial complex, denoted Γ(An)

The complex Γ(An)

We can identify elements of Sn+1 with faces a simplicial complex, denoted Γ(An) 5|17|36|24

157|36|24 5|1367|24 5|17|2346

13567|24 157|2346 5|123467 (∅ = 1234567)

• A facet of Γ(A6):

The Γ complex

Theorem (Nevo-P.)

There exists a simplicial complex Γ(∆) such that γ(∆) = f (Γ(∆)) for the following flag spheres:

The Γ complex

Theorem (Nevo-P.)

There exists a simplicial complex Γ(∆) such that γ(∆) = f (Γ(∆)) for the following flag spheres:

◮ ∆ is a Coxeter complex

The Γ complex

Theorem (Nevo-P.)

There exists a simplicial complex Γ(∆) such that γ(∆) = f (Γ(∆)) for the following flag spheres:

◮ ∆ is a Coxeter complex ◮ ∆ is (dual to) an associahedron

The Γ complex

Theorem (Nevo-P.)

There exists a simplicial complex Γ(∆) such that γ(∆) = f (Γ(∆)) for the following flag spheres:

◮ ∆ is a Coxeter complex ◮ ∆ is (dual to) an associahedron ◮ ∆ is (dual to) a cyclohedron

The Γ complex

Theorem (Nevo-P.)

There exists a simplicial complex Γ(∆) such that γ(∆) = f (Γ(∆)) for the following flag spheres:

◮ ∆ is a Coxeter complex ◮ ∆ is (dual to) an associahedron ◮ ∆ is (dual to) a cyclohedron ◮ ∆ has at most 2n+3 vertices

The Γ complex

Theorem (Nevo-P.)

There exists a simplicial complex Γ(∆) such that γ(∆) = f (Γ(∆)) for the following flag spheres:

◮ ∆ is a Coxeter complex ◮ ∆ is (dual to) an associahedron ◮ ∆ is (dual to) a cyclohedron ◮ ∆ has at most 2n+3 vertices ◮ [P.-Tenner] ∆ is a barycentric subdivision of with dim ∆ ≤ 8

(builds on Brenti-Welker)

Combinatorial interpretations for γ-vectors

Gal’s conjecture An example The Γ complex A conjecture

A conjecture - what γ counts?

Conjecture (Nevo-P.)

If ∆ is a flag homology sphere, then γ(∆) = f (Γ(∆)) for some (flag? balanced?) simplicial complex Γ(∆)

Questions?

“On γ-vectors satisfying the Kruskal-Katona inequalities,” with E. Nevo, Discrete and Computational Geometry, to appear. arXiv:0909.0694