Kazhdan-Lusztig polynomials of matroids

Transcription

1 Kazhdan-Lusztig polynomials of matroids Nicholas Proudfoot University of Oregon AMS Special Session on Arrangements of Hypersurfaces

2 Arrangements and Flats Let V be a finite dimensional vector space 1

3 Arrangements and Flats Let V be a finite dimensional vector space A a finite set of hyperplanes in V with H A H = {0} 1

4 Arrangements and Flats Let V be a finite dimensional vector space A a finite set of hyperplanes in V with H A F V a flat (intersection of some hyperplanes) H = {0} 1

5 Arrangements and Flats Let V be a finite dimensional vector space A a finite set of hyperplanes in V with H A F V a flat (intersection of some hyperplanes) H = {0} V F 1

6 Arrangements and Flats Definition The contraction of A at F is the arrangement A F := {H F F H A} in the vector space F. 2

7 Arrangements and Flats Definition The contraction of A at F is the arrangement A F := {H F F H A} in the vector space F. V F F A A F 2

8 Arrangements and Flats Definition The localization of A at F is the arrangement A F := {H/F F H A} in the vector space V /F. 3

9 Arrangements and Flats Definition The localization of A at F is the arrangement A F := {H/F F H A} in the vector space V /F. V V=F F A A F 3

10 Characteristic Polynomial Let χ A (t) Z[t] be the characteristic polynomial of A. 4

11 Characteristic Polynomial Let χ A (t) Z[t] be the characteristic polynomial of A. If V is a vector space over F q, χ A (q) = V H A H. 4

12 Characteristic Polynomial Let χ A (t) Z[t] be the characteristic polynomial of A. If V is a vector space over F q, χ A (q) = V H A H. Example V χ A (t) = t 2 3t + 2 4

13 Kazhdan-Lusztig Polynomial Theorem There exists a unique way to assign to each arrangement A a polynomial P A (t) Z[t] subject to the following conditions: 5

14 Kazhdan-Lusztig Polynomial Theorem There exists a unique way to assign to each arrangement A a polynomial P A (t) Z[t] subject to the following conditions: If dim V = 0, P A (t) = 1 5

15 Kazhdan-Lusztig Polynomial Theorem There exists a unique way to assign to each arrangement A a polynomial P A (t) Z[t] subject to the following conditions: If dim V = 0, P A (t) = 1 If dim V > 0, deg P A (t) < 1 2 dim V 5

16 Kazhdan-Lusztig Polynomial Theorem There exists a unique way to assign to each arrangement A a polynomial P A (t) Z[t] subject to the following conditions: If dim V = 0, P A (t) = 1 If dim V > 0, deg P A (t) < 1 2 dim V t dim V P A (t 1 ) = F χ AF (t)p A F (t). 5

17 Kazhdan-Lusztig Polynomial Theorem There exists a unique way to assign to each arrangement A a polynomial P A (t) Z[t] subject to the following conditions: If dim V = 0, P A (t) = 1 If dim V > 0, deg P A (t) < 1 2 dim V t dim V P A (t 1 ) = F χ AF (t)p A F (t). P A (t) is called the Kazhdan-Lusztig polynomial of A. 5

18 Kazhdan-Lusztig Polynomial Theorem There exists a unique way to assign to each arrangement A a polynomial P A (t) Z[t] subject to the following conditions: If dim V = 0, P A (t) = 1 If dim V > 0, deg P A (t) < 1 2 dim V t dim V P A (t 1 ) = F χ AF (t)p A F (t). P A (t) is called the Kazhdan-Lusztig polynomial of A. Remark The theory of Kazhdan-Lusztig-Stanley polynomials provides a common generalization of these polynomials and classical Kazhdan-Lusztig polynomials. 5

19 Geometric Interpretation V V /H = A 1 H A H A H A P 1 6

20 Geometric Interpretation V V /H = A 1 H A H A H A P 1 Definition We define the Schubert variety of A Y A := V H A P 1. 6

21 Geometric Interpretation V V /H = A 1 H A H A H A P 1 Definition We define the Schubert variety of A Y A := V H A P 1. Have H (Y A ) IH (Y A ), both concentrated in even degree. 6

22 Geometric Interpretation Theorem (Huh-Wang, P-Xu-Young, Elias-P-Wakefield) t i dim H 2i (Y A ) = F t codim F 7

23 Geometric Interpretation Theorem (Huh-Wang, P-Xu-Young, Elias-P-Wakefield) t i dim H 2i (Y A ) = F t i dim IH 2i (Y A ) = F t codim F t codim F P A F (t) =: Z A (t) 7

24 Geometric Interpretation Theorem (Huh-Wang, P-Xu-Young, Elias-P-Wakefield) t i dim H 2i (Y A ) = F t codim F t i dim IH 2i (Y A ) = t codim F P A F (t) =: Z A (t) F ( / ) t i dim IH 2i (Y A ) H 2 (Y A ) IH 2i 2 (Y A ) = P A (t) 7

25 Geometric Interpretation Theorem (Huh-Wang, P-Xu-Young, Elias-P-Wakefield) t i dim H 2i (Y A ) = F t codim F t i dim IH 2i (Y A ) = t codim F P A F (t) =: Z A (t) F ( / ) t i dim IH 2i (Y A ) H 2 (Y A ) IH 2i 2 (Y A ) = P A (t) Corollary The polynomial P A (t) has non-negative coefficients. 7

26 Geometric Interpretation Theorem (Huh-Wang, P-Xu-Young, Elias-P-Wakefield) t i dim H 2i (Y A ) = F t codim F t i dim IH 2i (Y A ) = t codim F P A F (t) =: Z A (t) F ( / ) t i dim IH 2i (Y A ) H 2 (Y A ) IH 2i 2 (Y A ) = P A (t) Corollary The polynomial P A (t) has non-negative coefficients. Remark The definition of P A (t) makes sense for matroids, but when the matroid is not realizable, non-negativity is still a conjecture. Work in progress by Braden-Huh-Matherne-P-Wang. 7

27 Example Let A n be an arrangement of n generic hyperplanes in C n 1. 8

28 Example Let A n be an arrangement of n generic hyperplanes in C n 1. What are the flats? 8

29 Example Let A n be an arrangement of n generic hyperplanes in C n 1. What are the flats? For each k < n 1, there are ( n k) flats of codimension k. 8

30 Example Let A n be an arrangement of n generic hyperplanes in C n 1. What are the flats? For each k < n 1, there are ( n k) flats of codimension k. For such a flat F, A F n = A n k 8

31 Example Let A n be an arrangement of n generic hyperplanes in C n 1. What are the flats? For each k < n 1, there are ( n k) flats of codimension k. For such a flat F, A F n = A n k and (A n ) F is Boolean of rank k. 8

32 Example Let A n be an arrangement of n generic hyperplanes in C n 1. What are the flats? For each k < n 1, there are ( n k) flats of codimension k. For such a flat F, A F n = A n k and (A n ) F is Boolean of rank k. There is a unique flat of codimension n 1. 8

33 Example Let A n be an arrangement of n generic hyperplanes in C n 1. What are the flats? For each k < n 1, there are ( n k) flats of codimension k. For such a flat F, A F n = A n k and (A n ) F is Boolean of rank k. There is a unique flat of codimension n 1. n 2 t n 1 P An (t 1 ) = k=0 n 2 = k=0 ( ) n (t 1) k P An k (t) + χ k An (t) ( ) n (t 1) k P An k (t) + (t 1)n + ( 1) n (t 1) k t 8

34 Example Put it in a generating function: Φ(t, u) := P An (t)u n 1 n=2 9

35 Example Put it in a generating function: Φ(t, u) := P An (t)u n 1 n=2 Then our recursion becomes Φ(t 1, tu) = t n 1 P An (t 1 )u n 1 = n=2 ( n 2 n=2 k=0 ( ) ) n (t 1) k P An k (t) + χ k An (t) u n 1 = ( ) u (1 + u) Φ t, 1 u(t 1) + u(t 1) (1 u(t 1)) = (1 + u) (1 u(t 1)) 2 9

36 Example Theorem (P-Wakefield-Young) Φ(t, u) = 2 u (2t + 1)u (2tu + 1)u + u 2 1 (2tu + 1) 2 10

37 Example Theorem (P-Wakefield-Young) Φ(t, u) = 2 u (2t + 1)u (2tu + 1)u + u 2 1 (2tu + 1) 2 P An (t) = ( )( ) 1 n n i 2 t i i + 1 i i i 0 10

38 Example Theorem (P-Wakefield-Young) Φ(t, u) = 2 u (2t + 1)u (2tu + 1)u + u 2 1 (2tu + 1) 2 P An (t) = ( )( ) 1 n n i 2 t i i + 1 i i i 0 Corollary (Gedeon-P-Young) The polynomial P An (t) is real-rooted for all n. 10

39 Example Theorem (P-Wakefield-Young) Φ(t, u) = 2 u (2t + 1)u (2tu + 1)u + u 2 1 (2tu + 1) 2 P An (t) = ( )( ) 1 n n i 2 t i i + 1 i i i 0 Corollary (Gedeon-P-Young) The polynomial P An (t) is real-rooted for all n. Remark The polynomial P A (t) is conjecturally real-rooted for any A, but this is the only non-trivial family of examples for which we can prove it! 10

40 Thanks! 11