Boolean Product Polynomials and the Resonance Arrangement

Transcription

1 Boolean Product Polynomials and the Resonance Arrangement Sara Billey University of Washington Based on joint work with: Lou Billera and Vasu Tewari FPSAC July 17, 2018

2 Outline Symmetric Polynomials Schur Positivity via GL n representation theory and vector bundles Corollaries and Generalizations (Work in Progress) Motivation

3 Symmetric Polynomials Notation. Fix an alphabet of variables X = {x 1, x 2,..., x n }. The symmetric group S n acts on C[x 1, x 2,..., x n ] by permuting the variables: w.x i = x w(i). A polynomial f C[x 1, x 2,..., x n ] is symmetric if w.f = f for all w S n. Let Λ n denote the ring of symmetric polynomials in C[x 1, x 2,..., x n ].

4 Symmetric Polynomials Examples. Let [n] = {1, 2,..., n}. Elementary: e k = Homogeneous: h k = Power sum: p k = A [n] A =k x i i A multisets A [n] A =k n xi k i=1 x i i A e 2 (x 1, x 2, x 3, x 4 ) = x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 p 2 (x 1, x 2, x 3, x 4 ) = x x x x 2 4 h 2 = e 2 + p 2.

5 Symmetric Polynomials Fact. Λ n = C[e 1,..., e n ] = C[h 1,..., h n ] = C[p 1,..., p n ]

6 Symmetric Polynomials Fact. Λ n = C[e 1,..., e n ] = C[h 1,..., h n ] = C[p 1,..., p n ] Question. What other symmetric polynomials are natural?

7 Symmetric Polynomials Fact. Λ n = C[e 1,..., e n ] = C[h 1,..., h n ] = C[p 1,..., p n ] Question. What other symmetric polynomials are natural? Monomials: m λ = x λ 1 1 x λ 2 2 x n λn + other monomials in S n -orbit Stanley s chromatic symmetric functions on a graph G = (V, E): X G (x 1,..., x n ) = c:v [n] v V proper coloring x c(v). Observe. These examples are all sums of products.

8 Schur Polynomials Defn. Given a partition λ = (λ 1,..., λ n ), the Schur polynomial s λ (x 1,..., x n ) = x i T SSYT (λ,n) i T where SSYT (λ, n) are the semistandard fillings of λ with positive integers in [n]. Semistandard implies strictly increasing in columns and leniently increasing in rows. Example. For λ = (2, 1) and n = 2, SSYT (λ, n) has two fillings so s (2,1) (x 1, x 2 ) = x 2 1 x 2 + x 1 x 2 2.

9 Boolean Product Polynomials Question. What about products of sums?

10 Boolean Product Polynomials Question. What about products of sums? Defn. For X = {x 1,..., x n }, define (n, k)-boolean Product Polynomial: For 1 k n, B n,k (X) := A [n] A =k x i i A n-th Total Boolean Product Polynomial: n B n (X) := B n,k (X) = x i k=1 A [n] i A A Example. B 2 = (x 1 )(x 2 )(x 1 + x 2 ) = x1 2x 2 + x 1 x2 2 = s (2,1)(x 1, x 2 )

11 Boolean Product Polynomials Examples. B 3,1 = (x 1 )(x 2 )(x 3 ) = e 3 (x 1, x 2, x 3 ) = s (1,1,1) (x 1, x 2, x 3 ) B 3,2 = (x 1 + x 2 )(x 1 + x 3 )(x 2 + x 3 ) = s (2,1) (x 1, x 2, x 3 ) B 3,3 = (x 1 + x 2 + x 3 ) = e 1 (x 1, x 2, x 3 ) = s (1) (x 1, x 2, x 3 ) B 3 = s (1,1,1) s (2,1) s (1) = s (4,2,1) + s (3,3,1) + s (3,2,2).

12 Subset Alphabets Defn. For 1 k n, define a new alphabet of linear forms X (k) = {x A = i A x i : A [n], A = k}. Then B n,k = A [n] A =k x A = e ( n k) (X (k) ).

13 Subset Alphabets Defn. For 1 k n, define a new alphabet of linear forms X (k) = {x A = i A x i : A [n], A = k}. Then B n,k = A [n] A =k x A = e ( n k) (X (k) ). Furthermore, for 1 p ( n k) define the symmetric polynomials e p (X (k) ) = S k-subsets of[n] S =p x A. A S

14 Schur Positivity Theorem. For all 1 k n and 1 p ( n k), the expansion e p (X (k) ) = λ c λ s λ (x 1,..., x n ) has nonnegative integer coefficients c λ. Corollary. Both B n,k and B n are Schur positive.

15 Proof Setup Notation. Fix a complex vector bundle E of rank n. The total Chern class c(e) is the sum of the individual Chern classes c(e) = 1 + c 1 (E) + + c n (E). Via the Splitting Principle, we have c(e) = n i=1 (1 + x i ) where the x i for 1 i n are the Chern roots of E associated to certain line bundles.

16 Prior Work Thm.(Lascoux, 1978) The total Chern class of 2 E and Sym 2 E is Schur-positive in terms of the Chern roots x 1,..., x n of E. Specifically, there exist integers d λ,µ 0 for µ λ such that c( 2 E) = c(sym 2 E) = 1 i<j n 1 i j n (1 + x i + x j ) = 2 (n 2) µ δ n 1 d γn,µ2 µ s µ (X), (1 + x i + x j ) = 2 (n 2) µ δ n d δn,µ2 µ s µ (X). Here γ n = (n 1,..., 1, 0) and δ n = (n,..., 2, 1).

17 Binomial Determinants Lascoux showed that for µ = (µ 1,..., µ n ) λ = (λ 1,..., λ n ), d λ,µ = det (( )) λi + n i 0. µ j + n j 1 i,j n

18 Binomial Determinants Lascoux showed that for µ = (µ 1,..., µ n ) λ = (λ 1,..., λ n ), d λ,µ = det (( )) λi + n i 0. µ j + n j 1 i,j n Thm.(Gessel-Viennot 1985) d λ,µ counts the number of nonintersecting lattice paths from heights λ + δ n along the y-axis to main diagonal points µ + δ n using east or south steps. This highly influential theorem was inspired by Lascoux s theorem!

19 Vector Bundle Approach to Schur Positivity Notation. Fix a complex vector bundle E of rank n over a smooth projective variety V. The total Chern class n c(e) = 1 + c 1 (E) + + c n (E) = (1 + x i ) i=1 where the x i for 1 i n are the Chern roots of E. Construct another vector bundle S λ (E) over V by applying the Schur functor from GL n -representation theory on each fiber. Thm.(Fulton) The Chern roots of S λ (E) are indexed by semistandard tableaux: {x T = x i for T SSYT (λ, n)}. i T

20 Vector Bundle Approach to Schur Positivity Notation. For any partitions λ and µ, consider the Schur function s µ on the alphabet of Chern roots on S λ (E), denoted s µ (S λ (E)). Example. Take n = 3, λ = (1, 1), then the Chern roots of S λ (E) are the variables in the alphabet For µ = (2, 1), expand X (2) = {x 1 + x 2, x 1 + x 3, x 2 + x 3 }. s µ (S λ (E)) =s (2,1) (x 1 + x 2, x 1 + x 3, x 2 + x 3 ) =2s (3) (x 1, x 2, x 3 ) + 5s (2,1) (x 1, x 2, x 3 ) + 4s (1,1,1) (x 1, x 2, x 3 ).

21 Vector Bundle Approach to Schur Positivity Notation. For any partitions λ and µ, consider the Schur function s µ on the alphabet of Chern roots on S λ (E), denoted s µ (S λ (E)). Example. Take n = 3, λ = (1, 1), then the Chern roots of S λ (E) are the variables in the alphabet For µ = (2, 1), expand X (2) = {x 1 + x 2, x 1 + x 3, x 2 + x 3 }. s µ (S λ (E)) =s (2,1) (x 1 + x 2, x 1 + x 3, x 2 + x 3 ) =2s (3) (x 1, x 2, x 3 ) + 5s (2,1) (x 1, x 2, x 3 ) + 4s (1,1,1) (x 1, x 2, x 3 ). Note. This operation is not plethysm s µ [s λ ].

22 Key Ingredient Thm.(Pragacz 1996) Let λ be a partition, and let E 1,..., E k be vector bundles, Y 1,..., Y k be the alphabets consisting of their Chern roots, µ (1),..., µ (k) be partitions. Then, there exists nonnegative integers c (ν (1),...,ν (k) ) such that s λ (S µ(1) (E 1 ) S µ(k) (E k )) = c (ν (1),...,ν (k) ) s ν 1 (Y 1 ) s νk (Y k ). ν 1,...,ν k Pragacz s proof uses work of Fulton-Lazarsfeld on numerical positivity for ample vector bundles. The Hard Lefschetz theorem is a key component.

23 Corollaries Cor. For any partitions λ, µ, s µ (S λ (E)) is Schur positive. Cor. The expansion of e p (X (k) ) is Schur positive since the Chern roots of S 1k (E) = X (k) and e p = s 1 p. Cor. The analogue of Lascoux s theorem holds for all Schur functors S λ (E)), e.g. c( k E) = A [n], A =k 1 + x i = e p (X (k) ). i A p 0 Question. What are the Schur expansions?

24 Boolean Product Expansions for B n,n 1 Thm. For n 2, n B n,n 1 = 1 + x x n x i ) = i=1(x a λ s λ (X) λ n where a λ is the number of T SYT (λ) with smallest ascent given by an even number. More generally, consider n B n,n 1 (X; q) := (h 1 (X) + qx i ) i=1 n = q j e j (X)h (1 n j )(X). j=0 **New**: Brendon Rhoades has a new graded S n -module with B n,n 1 (X; q) as the graded Frobenius characteristic.

25 Motivation Defn. Consider the real variety V (B n ). Since each factor of B n is linear, this variety is a hyperplane arrangement called the Resonance Arrangement or All-Subsets Arrangement H n. Each hyperplane is orthogonal to a nonzero 0-1-vector in R n. Open. Find the characteristic polynomial for H n. Use it to count the number of regions and number of bounded regions by Zaslavsky s Theorem. Thm.(Cavalieri-Johnson-Markwig, 2011) The regions of H n are the domains of polynomiality of double Hurwitz numbers.

26 Motivation Further Connections. See Lou Billera s talk slides On the real linear algebra of vectors of zeros and ones 1. The chambers of the Resonance Arrangement H n can be labeled by maximal unbalanced collections of 0-1 vectors. See Billera-Moore-Moraites-Wang-Williams, Minimal balanced collections determine the minimum linear description of cooperative games possessing a nonempty core in Lloyd Shapley s economic game theory work from Finding a good formula for enumerating them is still open. 3. Hadamard s maximal determinant problem from 1893 can be rephrased in terms of finding maximal absolute value determinants of 0-1 matrices.

27 Many Thanks!