Outside nested decompositions and Schur function determinants. Emma Yu Jin

Transcription

1 Outside nested decompositions and Schur function determinants Emma Yu Jin Technische Universität Wien 77th SLC, Strobl September 12,

2 2 (Semi)standard Young tableaux

3 (Semi)standard Young tableaux (SSYT/SYT) λ/µ = (6, 6, 6, 4)/(3, 1) SSYT SYT strictly strictly < < weakly strictly < 3

4 Schur function s λ/µ (X ) and f λ/µ s λ/µ (X ) is the generating function of SSYT. s (2,2)/(1) (X ) = x1 2x 2 + x 1 x2 2 + x 1 2x 3+x 1 x x 1x 2 x 3 + = the number f λ/µ of SYT with entries from 1 to λ/µ is f (2,2)/(1) = [x 1 x 2 x 3 ]s (2,2)/(1) (X ) = 2. 4

5 5 Determinantal formulas for s λ/µ (X )

6 s (X ) = det s (X ) s (X ) s (X ) s (X ) s s Jacobi-Trudi determinant (X ) = det s s (X ) (X ) s (X ) Dual Jacobi-Trudi determinant s (X ) = det (X ) s (X ) s (X ) s (X ) s (X ) Giambelli determinant 6

7 s (X ) = det s (X ) s (X ) s (X ) s (X ) s s Jacobi-Trudi determinant (X ) = det s s (X ) (X ) s (X ) Dual Jacobi-Trudi determinant s (X ) = det (X ) s (X ) s (X ) s (X ) s (X ) Giambelli determinant 7

8 s (X ) = det s (X ) s (X ) s (X ) s (X ) s s Jacobi-Trudi determinant (X ) = det s s (X ) (X ) s (X ) Dual Jacobi-Trudi determinant s (X ) = det (X ) s (X ) s (X ) s (X ) s (X ) Giambelli determinant 8

9 Jacobi-Trudi determinant and its dual s λ/µ (X ), Giambelli determinant s λ (X ), Lascoux and Pragacz determinant s λ/µ (X ), Hamel and Goulden determinant s λ/µ (X ), (outside decompositions) unified by Reference: A.M. Hamel and I.P. Goulden, Planar decompositions of tableaux and Schur function determinants, Europ. J. Combinatorics, 16, , (border) strip or ribbon are not allowed 9

10 10 Hamel and Goulden s Determinant

11 Motivation: unify different determinantal expressions of s λ/µ (X ) Hamel and Goulden determinant, (outside decompositions) If the skew diagram of λ/µ is edgewise connected. Then, for any outside decomposition Φ = (θ 1, θ 2,..., θ g ) of the skew shape λ/µ, we have s λ/µ (X ) = det[s θi #θ j (X )] g i,j=1 where s (X ) = 1 and s θi #θ j (X ) = 0 if θ i #θ j is undefined. 11

12 (Border) strips (or ribbons): A skew diagram θ is a (border) strip if θ is edgewise connected and contains no 2 2 blocks of boxes. Yes No No 12

13 φ = (θ 1, θ 2, θ 3, θ 4 ) is an outside decomposition (1) θ i is a (border) strip for all i; (2) the disjoint union of all (border) strips is the skew shape λ/µ; (3) every starting box (resp. ending box) of θ i is on the bottom or left (resp. the top or right) perimeter of the skew shape λ/µ. θ 2 θ 1 θ 3 θ 4 Yes 13

14 φ = (θ 1, θ 2, θ 3, θ 4, θ 5 ) is not an outside decomposition (1) θ i is a (border) strip for all i; (2) the disjoint union of all (border) strips is the skew shape λ/µ; (3) every starting box (resp. ending box) of θ i is on the bottom or left (resp. the top or right) perimeter of the skew shape λ/µ. θ 2 θ 5 θ 1 No θ 3 θ 4 14

15 Outside decomposition (is nested) cutting strip operator # θ 2 θ 1 θ 3 θ 4 E.g., θ 4 #θ 3 = (3, 3)/(2) and θ 3 #θ 4 is undefined. Simplify the definition of θ i #θ j from Hamel and Goulden s paper: W.Y.C. Chen, G.G Yan and A.L.B Yang, Transformations of border strips and Schur function determinants, J. Algebr. Comb. 21, ,

16 Hamel and Goulden determinant, (outside decompositions) If the skew diagram of λ/µ is edgewise connected. Then, for any outside decomposition Φ = (θ 1, θ 2,..., θ g ) of the skew shape λ/µ, we have s λ/µ (X ) = det[s θi #θ j (X )] g i,j=1, where s (X ) = 1 and s θi #θ j (X ) = 0 if θ i #θ j is undefined. 16

17 Jacobi-Trudi determinant Dual Jacobi-Trudi determinant Lascoux-Pragacz determinant Giambelli determinant 17

18 Positive side: simplify some determinants q(θ 1 ) θ 2 θ 1 cutting strip q(θ 2 ) p(θ 1 ) s λ/µ (X ) = det p(θ 2 ) [ sθ1 (X ) s θ1 #θ 2 (X ) s θ2 #θ 1 (X ) s θ2 (X ) ] 18

19 Negative side: can not simplify some determinants # minimal strips= # columns # minimal strips= # rows 19

20 Main results (outside nested decompositions) thickened strip is allowed are not allowed 20

21 Main results: a determinantal expression of s λ/µ (X )p 1 r (X ) (outside nested decompositions) If the skew diagram of λ/µ is edgewise connected. Then, for any outside nested decomposition Φ = (Θ 1, Θ 2,..., Θ g ) of the skew shape λ/µ, we have p 1 r (X )s λ/µ (X ) = det[s Θi #Θ j (X )] g i,j=1 where p 1 r (X ) = ( x i ) r, s (X ) = 1, s Θi #Θ j (X ) = 0 if Θ i #Θ j is undefined and r is the number of common special corners of Φ. i=1 21

22 Main results: a determinantal expression of s λ/µ (X )p 1 r (X ) (outside nested decompositions) + exponential specialization If the skew diagram of λ/µ is edgewise connected. Then, for any outside nested decomposition Φ = (Θ 1, Θ 2,..., Θ g ) of the skew shape λ/µ, we have f λ/µ = λ/µ! det[( Θ i #Θ j!) 1 f Θ i #Θ j ] g i,j=1 where f = 1 and f Θ i #Θ j = 0 if Θ i #Θ j is undefined. 22

23 Outside nested decomposition: p 1 4(X )s (6,6,6,4)/(3,1) (X ) = det 23

24 Application to the m-strip tableaux: Reference: Y. Baryshnikov and D. Romik, Enumeration formulas for Young tableaux in a diagonal strip, Israel Journal of Mathematics 178, , an outside nested decomposition of an m-strip diagram 24

25 25 Outside nested decompositions

26 Thickened strips: A skew diagram Θ is a thickened strip if Θ is edgewise connected and neither contains a 3 2 block of boxes nor a 2 3 block of boxes. Yes No No 26

27 Φ = (Θ 1, Θ 2 ) is an outside thickened strip decomposition. Θ 2 Θ 1 (1) Θ i is a thickened strip for all i. (2) the union of all thickened strips is the skew shape λ/µ. (3) every starting box (resp. ending box) of Θ i is on the bottom or left (resp. the top or right) perimeter of the skew shape λ/µ. (4) allowed common special corners (next page) 27

28 Special corners: and Special upper corners: Special lower corners: 28

29 Outside thickened strip decompositions: allowed common special corners 29

30 Non-outside thickened strip decomposition: NOT allowed common special corners Θ 2 Θ 3 Θ 1 is not an outside thickened strip decomposition. 30

31 Outside nested decompositions: Θ 2 Θ 2 Θ 1 Θ 1 Yes No for all c, all boxes of content c all go up or all go right; or all boxes of content c are all special corners; or all boxes of content (c + 1) are all special corners. 31

32 Thickened cutting strip H(Φ): Θ 2 Θ 1 H(Φ): 32

33 Define Θ i #Θ j = [p(θ j ), q(θ i )] q(θ 1 ) p(θ 1 ) H(Φ): q(θ 2 ) Θ 2 Θ 1 p 1 4(X )s (6,6,6,4)/(3,1) (X ) = det p(θ 2 ) [ sθ1 (X ) s Θ1 #Θ 2 (X ) s Θ2 #Θ 1 (X ) s Θ2 (X ) ] 33

34 Define Θ i #Θ j = [p(θ j ), q(θ i )] q(θ 1 ) p(θ 1 ) H(Φ): q(θ 2 ) Θ 2 Θ 1 p(θ 2 ) p 1 4(X )s (6,6,6,4)/(3,1) (X ) = det 34

35 Define Θ i #Θ j = [p(θ j ), q(θ i )] q(θ 1 ) p(θ 1 ) H(Φ): q(θ 2 ) Θ 2 Θ 1 p(θ 2 ) p 1 4(X )s (6,6,6,4)/(3,1) (X ) = det 35

36 Define Θ i #Θ j = [p(θ j ), q(θ i )] q(θ 1 ) p(θ 1 ) H(Φ): q(θ 2 ) Θ 2 Θ 1 p(θ 2 ) f (6,6,6,4)/(3,1) = (18)! det f 1 (11)! f 1 (11)! f 1 (11)! f 1 (11)! 36

37 37 Proof of the main results:

38 The proof consists of three main steps: (1) SSYT a sequence of non-crossing double lattice paths based on the bijection between SSYT and a sequence of non-intersecting lattice paths in Hamel and Goulden s paper. (2) Define a sequence of separable double lattice paths, whose generating function is p 1 r (X )s λ/µ (X ). (3) Construct an involution on all non-separable sequences of double lattice paths, so that only the separable ones constribute the determinant det[s Θi #Θ j (X )]. [Reference: J.R. Stembridge, Nonintersecting paths, pfaffians and plane partitions, Adv. Math., 83, , 1990.] 38

39 Applications an m-strip diagram n m is fixed and small m 39

40 Examples: a 5-strip diagram has 8 strips has 2 thickened strips 40

41 Counting 3-strip tableaux D 3n 2 D 3n 1 n columns (3n 2) boxes n columns (3n 1) boxes To prove (3n 1)f D 3n 2 = 2f D 3n 1 41

42 Counting 3-strip tableaux To prove (3n 1)f D 3n 2 a +i = 2f D 3n 1 a i SYT(D 3n 2 ) +i i a a i x = 3n x a i 42

43 Counting 3-strip tableaux D 3n 2 D 3n 2,i To prove remove box (i, n i) (3n 2)f D 3n 2,i = f D 3n 2 + f D 3n 2 = n 1 i=1 f D 3n 2,i. ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 43

44 (3n 2)f D 3n 2,i = f D 3n 2 + ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 a b c +r r a b c if r < min{a, b} SYT(D 3n 2,i ) 44

45 (3n 2)f D 3n 2,i = f D 3n 2 + ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 a c +r a b c if r > min{a, b} = a SYT(D 3n 2,i ) = a r b c 45

46 (3n 2)f D 3n 2,i = f D 3n 2 + ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 a c +r a b c if r > min{a, b} = a SYT(D 3n 2,i ) = a r b c SYT(D 3n 3i 1 ) SYT(D 3i 1 ) 46

47 (3n 2)f D 3n 2,i = f D 3n 2 + ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 a c +r a b c if r > min{a, b} = b SYT(D 3n 2,i ) = a c b r 47

48 (3n 2)f D 3n 2,i = f D 3n 2 + ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 a c +r a b c if r > min{a, b} = b SYT(D 3n 2,i ) = a c b r SYT(D 3n 3i 1 ) SYT(D 3i 1 ) 48

49 (3n 1)f D 3n 2 = 2f D 3n 1. (3n 2)f D 3n 2,i = f D 3n 2 + ( ) 3n 2 f D 3i 1 f D 3n 3i 1. 3i 1 g.f. = f (x) = 2g(x), f (x) = 1 + g(x) 2. = f (x) = 2 tan(x/2), g(x) = tan(x/2). where f (x) = n 1 f D 3n 2 (3n 2)! x 2n 1, g(x) = n 1 f D 3n 1 (3n 1)! x 2n 1. 49

50 Counting 3-strip tableaux D 3n 2 D 3n 1 f D 3n 2 = (3n 2)!E 2n 1 (2n 1)!2 2n 2, f D 3n 1 = (3n 1)!E 2n 1 (2n 1)!2 2n 1. 50

51 Summary: Jacobi-Trudi determinant and its dual s λ/µ (X ), Giambelli determinant s λ (X ), Lascoux and Pragacz determinant s λ/µ (X ), Hamel and Goulden determinant s λ/µ (X ), (outside decompositions) Jin, 2016, a determinant p 1 r (X )s λ/µ (X ), (outside nested decompositions) unified by generalized by to count m-strip tableaux. 51

52 52 Vielen Dank!