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, f k (∆) = number of faces of dimension k − 1 n � f k (∆) t k f (∆; t ) := k =0 ( f 0 , f 1 , . . . , f n ) is the f -vector n � h (∆; t ) := (1 − t ) n f (∆; t / (1 − t )) = h k (∆) t k k =0 ( h 0 , h 1 , . . . , h n ) is the h-vector

The f - and h -vectors Let ∆ be an ( n − 1)-dimensional simplicial complex, f k (∆) = number of faces of dimension k − 1 n � f k (∆) t k f (∆; t ) := k =0 ( f 0 , f 1 , . . . , f n ) is the f -vector n � h (∆; t ) := (1 − t ) n f (∆; t / (1 − t )) = h k (∆) t k k =0 ( h 0 , h 1 , . . . , h n ) is the h-vector f -vectors are characterized by the Kruskal-Katona inequalities

The f - and h -vectors • • • • ∆ : • •

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6 ◮ f 2 = 6

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6 ◮ f 2 = 6 f (∆; t ) = 1 + 6 t + 6 t 2

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6 ◮ f 2 = 6 f (∆; t ) = 1 + 6 t + 6 t 2 = (1 + 2 t + t 2 )

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6 ◮ f 2 = 6 f (∆; t ) = 1 + 6 t + 6 t 2 = (1 + 2 t + t 2 ) + 4 t (1 + t )

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6 ◮ f 2 = 6 f (∆; t ) = 1 + 6 t + 6 t 2 = (1 + 2 t + t 2 ) + 4 t (1 + t ) + t 2 (1)

The f - and h -vectors • • • • ∆ : • • ◮ f 0 = 1 ◮ f 1 = 6 ◮ f 2 = 6 f (∆; t ) = 1 + 6 t + 6 t 2 = (1 + 2 t + t 2 ) + 4 t (1 + t ) + t 2 (1) h (∆; t ) = 1 + 4 t + t 2

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0 e.g., 1 + 3 t + 7 t 2 + 3 t 3 + t 4

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0 e.g., 1 + 3 t + 7 t 2 + 3 t 3 + t 4 1 + 4 t + 6 t 2 + 4 t 3 + t 4

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0 e.g., 1 + 3 t + 7 t 2 + 3 t 3 + t 4 1 + 4 t + 6 t 2 + 4 t 3 + t 4 − t (1 + 2 t + t 2 )

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0 e.g., 1 + 3 t + 7 t 2 + 3 t 3 + t 4 1 + 4 t + 6 t 2 + 4 t 3 + t 4 − t (1 + 2 t + t 2 ) +3 t 2

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0 e.g., 1 + 3 t + 7 t 2 + 3 t 3 + t 4 1 + 4 t + 6 t 2 + 4 t 3 + t 4 − t (1 + 2 t + t 2 ) +3 t 2 1 + 3 t + 7 t 2 + 3 t 3 + t 4 = (1 + t ) 4 − t (1 + t ) 2 + 3 t 2

The γ -vector If h ( t ) = � n i =0 h i t i is symmetric, then there exist γ i such that n / 2 � γ i t i (1 + t ) n − 2 i , h ( t ) = i =0 e.g., 1 + 3 t + 7 t 2 + 3 t 3 + t 4 1 + 4 t + 6 t 2 + 4 t 3 + t 4 − t (1 + 2 t + t 2 ) +3 t 2 1 + 3 t + 7 t 2 + 3 t 3 + t 4 = (1 + t ) 4 − t (1 + t ) 2 + 3 t 2 the vector ( γ 0 , γ 1 , . . . ) is called the γ -vector

Gal’s conjecture For ∆ a sphere, Dehn-Sommerville relations say h (∆) is symmetric, h i = h n − i , and hence ∆ has a well-defined γ -vector, denoted γ (∆)

Gal’s conjecture For ∆ a sphere, Dehn-Sommerville relations say h (∆) is symmetric, h i = h n − 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, h i = h n − 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, h i = h n − 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, h i = h n − 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

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Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 132 213 231 312 321

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 213 231 312 321

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 1 t 213 231 312 321

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 1 t 213 1 t 231 312 321

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 1 t 213 1 t 231 1 t 312 321

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 1 t 213 1 t 231 1 t 312 1 t 321

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 1 t 213 1 t 231 1 t 312 1 t t 2 321 2

Eulerian polynomials Let w ∈ S n +1 and d ( w ) := |{ i : w i > w i +1 }| . Then, � t d ( w ) = h (∆( A n ); t ) A n ( t ) = w ∈ S n +1 t d ( w ) w d ( w ) 123 0 1 132 1 t 213 1 t 231 1 t 312 1 t t 2 321 2 A 2 ( t ) = 1 + 4 t + t 2 = h (∆( A 2 ); t )

Eulerian polynomials We have: A 1 ( t ) = 1 + t A 2 ( t ) = 1 + 4 t + t 2 A 3 ( t ) = 1 + 11 t + 11 t 2 + t 3 A 4 ( t ) = 1 + 26 t + 66 t 2 + 26 t 3 + t 4 . . .

Eulerian polynomials We have: A 1 ( t ) = 1 + t = (1 + t ) A 2 ( t ) = 1 + 4 t + t 2 A 3 ( t ) = 1 + 11 t + 11 t 2 + t 3 A 4 ( t ) = 1 + 26 t + 66 t 2 + 26 t 3 + t 4 . . .

Eulerian polynomials We have: A 1 ( t ) = 1 + t = (1 + t ) A 2 ( t ) = 1 + 4 t + t 2 = (1 + t ) 2 + 2 t A 3 ( t ) = 1 + 11 t + 11 t 2 + t 3 A 4 ( t ) = 1 + 26 t + 66 t 2 + 26 t 3 + t 4 . . .

Eulerian polynomials We have: A 1 ( t ) = 1 + t = (1 + t ) A 2 ( t ) = 1 + 4 t + t 2 = (1 + t ) 2 + 2 t A 3 ( t ) = 1 + 11 t + 11 t 2 + t 3 = (1 + t ) 3 + 8 t (1 + t ) A 4 ( t ) = 1 + 26 t + 66 t 2 + 26 t 3 + t 4 . . .

Eulerian polynomials We have: A 1 ( t ) = 1 + t = (1 + t ) A 2 ( t ) = 1 + 4 t + t 2 = (1 + t ) 2 + 2 t A 3 ( t ) = 1 + 11 t + 11 t 2 + t 3 = (1 + t ) 3 + 8 t (1 + t ) A 4 ( t ) = 1 + 26 t + 66 t 2 + 26 t 3 + t 4 = (1 + t ) 4 + 22 t (1 + t ) 2 + 16 t 2 . . .

The γ -vector for A n Define � S n = { w ∈ S n : w n − 1 < w n , and if w i − 1 > w i then w i < w i − 1 }

The γ -vector for A n Define � S n = { w ∈ S n : w n − 1 < w n , and if w i − 1 > w i then w i < w i − 1 } Theorem (Foata-Sch¨ utzenberger (1970)) � t d ( w ) (1 + t ) n − 2 d ( w ) , A n ( t ) = w ∈ b S n +1

The γ -vector for A n Define � S n = { w ∈ S n : w n − 1 < w n , and if w i − 1 > w i then w i < w i − 1 } Theorem (Foata-Sch¨ utzenberger (1970)) � t d ( w ) (1 + t ) n − 2 d ( w ) , A n ( t ) = w ∈ b S n +1 i.e., γ i ( A n ) = |{ w ∈ � S n +1 : d ( w ) = i }|

The γ -vector for A n � t d ( w ) (1 + t ) n − 2 d ( w ) A n ( t ) = w ∈ b S n +1