Puzzles Littlewood-Richardson coefficients and Horn inequalities

Transcription

1 Puzzles Littlewood-Richardson coefficients and Horn inequalities Olga Azenhas CMUC, Centre for Mathematics, University of Coimbra Seminar of the Mathematics PhD Program UCoimbra-UPorto Porto, 6 October 009 / 9

2 Littlewood-Richardson (LR) coefficients c λ µ ν are non-negative integer numbers depending on three non-negative integer vectors µ, ν, λ ordered decreasingly. / 9

3 Littlewood-Richardson (LR) coefficients c λ µ ν are non-negative integer numbers depending on three non-negative integer vectors µ, ν, λ ordered decreasingly. Who cares? 3 / 9

4 Littlewood-Richardson (LR) coefficients c λ µ ν are non-negative integer numbers depending on three non-negative integer vectors µ, ν, λ ordered decreasingly. Who cares? How can one evaluate them? 4 / 9

5 Littlewood-Richardson (LR) coefficients c λ µ ν are non-negative integer numbers depending on three non-negative integer vectors µ, ν, λ ordered decreasingly. Who cares? How can one evaluate them? What do they count? 5 / 9

6 Littlewood-Richardson (LR) coefficients c λ µ ν are non-negative integer numbers depending on three non-negative integer vectors µ, ν, λ ordered decreasingly. Who cares? How can one evaluate them? What do they count? Are there conditions to see whether or not a given LR-coefficient is non-zero? 6 / 9

7 Littlewood-Richardson coefficients: c λ µ ν Schur functions {s λ } λ form a Z-basis for the ring of symmetric functions s µ s ν = X λ c λ µ νs λ. 7 / 9

8 Littlewood-Richardson coefficients: c λ µ ν Schur functions {s λ } λ form a Z-basis for the ring of symmetric functions s µ s ν = X λ c λ µ νs λ. For which λ does s µ s ν contain s λ as a (positive) summand? 8 / 9

9 Littlewood-Richardson coefficients: c λ µ ν Schur functions {s λ } λ form a Z-basis for the ring of symmetric functions s µ s ν = X λ c λ µ νs λ. For which λ does s µ s ν contain s λ as a (positive) summand? Given µ, ν and λ when does one have c λ µ ν > 0? 9 / 9

10 Littlewood-Richardson coefficients: c λ µ ν Schur functions {s λ } λ form a Z-basis for the ring of symmetric functions s µ s ν = X λ c λ µ νs λ. For which λ does s µ s ν contain s λ as a (positive) summand? Given µ, ν and λ when does one have c λ µ ν > 0? The tensor product of two irreducible polynomial representations V µ and V ν of the general linear group GL d (C) decomposes into irreducible representations of GL d (C) V µ V ν = X l(λ) d c λ µ νv λ. 0 / 9

11 Littlewood-Richardson coefficients: c λ µ ν Schur functions {s λ } λ form a Z-basis for the ring of symmetric functions s µ s ν = X λ c λ µ νs λ. For which λ does s µ s ν contain s λ as a (positive) summand? Given µ, ν and λ when does one have c λ µ ν > 0? The tensor product of two irreducible polynomial representations V µ and V ν of the general linear group GL d (C) decomposes into irreducible representations of GL d (C) V µ V ν = X l(λ) d c λ µ νv λ. / 9

12 Littlewood-Richardson coefficients: c λ µ ν Schur functions {s λ } λ form a Z-basis for the ring of symmetric functions s µ s ν = X λ c λ µ νs λ. For which λ does s µ s ν contain s λ as a (positive) summand? Given µ, ν and λ when does one have c λ µ ν > 0? The tensor product of two irreducible polynomial representations V µ and V ν of the general linear group GL d (C) decomposes into irreducible representations of GL d (C) V µ V ν = X l(λ) d c λ µ νv λ. Given µ and ν, for which λ does V λ appear (with positive multiplicity) in V µ V ν? Given µ, ν and λ when does one have c λ µ ν > 0? / 9

13 Littlewood-Richardson coefficients: c λ µ ν 3 / 9

14 Littlewood-Richardson coefficients: c λ µ ν Schubert classes σ λ form a linear basis for H (G(d, n)), the cohomology ring of the Grassmannian G(d, n) of complex d-dimensional linear subspaces of C n, σ µ σ ν = X λ d (n d) c λ µ νσ λ. 4 / 9

15 Littlewood-Richardson coefficients: c λ µ ν Schubert classes σ λ form a linear basis for H (G(d, n)), the cohomology ring of the Grassmannian G(d, n) of complex d-dimensional linear subspaces of C n, σ µ σ ν = X λ d (n d) c λ µ νσ λ. There exist n n non singular matrices A, B and C, over a local principal ideal domain, with Smith invariants µ = (µ,..., µ n ), ν = (ν,..., ν n ) and λ = (λ,..., λ n ) respectively, such that AB = C if and only if c λ µ ν > 0. 5 / 9

16 Littlewood-Richardson coefficients: c λ µ ν Schubert classes σ λ form a linear basis for H (G(d, n)), the cohomology ring of the Grassmannian G(d, n) of complex d-dimensional linear subspaces of C n, σ µ σ ν = X λ d (n d) c λ µ νσ λ. There exist n n non singular matrices A, B and C, over a local principal ideal domain, with Smith invariants µ = (µ,..., µ n ), ν = (ν,..., ν n ) and λ = (λ,..., λ n ) respectively, such that AB = C if and only if c λ µ ν > 0. There exist n n Hermitian matrices A, B and C, with integer eigenvalues arranged in weakly decreasing order µ = (µ,..., µ n ), ν = (ν,..., ν n ) and λ = (λ,..., λ n ) respectively, such that C = A + B if and only if c λ µ,ν > 0. 6 / 9

17 . Schur functions Partitions and Young diagrams Fix a positive integer r. 7 / 9

18 . Schur functions Partitions and Young diagrams Fix a positive integer r. λ = (λ,..., λ r ), with λ λ r > 0 positive integers, is a partition of length l(λ) = r. 8 / 9

19 . Schur functions Partitions and Young diagrams Fix a positive integer r. λ = (λ,..., λ r ), with λ λ r > 0 positive integers, is a partition of length l(λ) = r. Each partition λ is identified with a Young (Ferrer) diagram λ 9 / 9

20 . Schur functions Partitions and Young diagrams Fix a positive integer r. λ = (λ,..., λ r ), with λ λ r > 0 positive integers, is a partition of length l(λ) = r. Each partition λ is identified with a Young (Ferrer) diagram λ consisting of λ = λ + + λ r boxes arranged in r bottom left adjusted rows of lengths λ λ r > 0. Example λ = (4, 3, ), λ = 9, l(λ) = 3 λ = 0 / 9

21 Young Tableaux n r, λ = (λ,..., λ r ), l(λ) = r. A semistandard tableau T of shape λ is a filling of the boxes of the Ferrer diagram λ with elements i in {,..., n} which is weakly increasing across rows from left to right strictly increasing up columns / 9

22 Young Tableaux n r, λ = (λ,..., λ r ), l(λ) = r. A semistandard tableau T of shape λ is a filling of the boxes of the Ferrer diagram λ with elements i in {,..., n} which is Example weakly increasing across rows from left to right strictly increasing up columns T has type α = (α,..., α n ) if T has α i entries equal i. λ = (4, 3, ), l(λ) = 3, n = 6 / 9

23 Young Tableaux n r, λ = (λ,..., λ r ), l(λ) = r. A semistandard tableau T of shape λ is a filling of the boxes of the Ferrer diagram λ with elements i in {,..., n} which is Example weakly increasing across rows from left to right strictly increasing up columns T has type α = (α,..., α n ) if T has α i entries equal i. λ = (4, 3, ), l(λ) = 3, n = 6 T = / 9

24 Young Tableaux n r, λ = (λ,..., λ r ), l(λ) = r. A semistandard tableau T of shape λ is a filling of the boxes of the Ferrer diagram λ with elements i in {,..., n} which is Example weakly increasing across rows from left to right strictly increasing up columns T has type α = (α,..., α n ) if T has α i entries equal i. λ = (4, 3, ), l(λ) = 3, n = 6 T = semistandard tableau T of shape λ = (4, 3, ), 4 / 9

25 Young Tableaux n r, λ = (λ,..., λ r ), l(λ) = r. A semistandard tableau T of shape λ is a filling of the boxes of the Ferrer diagram λ with elements i in {,..., n} which is Example weakly increasing across rows from left to right strictly increasing up columns T has type α = (α,..., α n ) if T has α i entries equal i. λ = (4, 3, ), l(λ) = 3, n = 6 T = semistandard tableau T of shape λ = (4, 3, ), α = (0,,, 3,, 3). 5 / 9

26 Young Tableaux n r, λ = (λ,..., λ r ), l(λ) = r. A semistandard tableau T of shape λ is a filling of the boxes of the Ferrer diagram λ with elements i in {,..., n} which is Example weakly increasing across rows from left to right strictly increasing up columns T has type α = (α,..., α n ) if T has α i entries equal i. λ = (4, 3, ), l(λ) = 3, n = 6 T = semistandard tableau T of shape λ = (4, 3, ), α = (0,,, 3,, 3). 6 / 9

27 Schur functions Example n = 7 T = x α(t ) = x 0 x x 3 x 3 4 x 5x 3 6 x 0 7 α(t )=(0,,,3,,3,0) 7 / 9

28 Schur functions continued Let x = (x,..., x n ) be a sequence of variables. 8 / 9

29 Schur functions continued Let x = (x,..., x n ) be a sequence of variables. Given the partition λ, the Schur function (polynomial) s λ (x) associated with the partition λ is the homogeneous polynomial of degree λ on the variables x..., x n X s λ (x) = T where T runs over all semistandard tableaux of shape λ on the alphabet {,..., n}. Example λ = (, ), λ = 3 n = 3 X α(t ) / 9

30 Schur functions continued Let x = (x,..., x n ) be a sequence of variables. Given the partition λ, the Schur function (polynomial) s λ (x) associated with the partition λ is the homogeneous polynomial of degree λ on the variables x..., x n X s λ (x) = T where T runs over all semistandard tableaux of shape λ on the alphabet {,..., n}. Example λ = (, ), λ = 3 n = 3 X α(t ) s λ (x, x, x 3 ) = x x + x x 3 + x x + x x x 3 + x x 3 + x x 3 + x x / 9

31 Schur functions continued Let x = (x,..., x n ) be a sequence of variables. Given the partition λ, the Schur function (polynomial) s λ (x) associated with the partition λ is the homogeneous polynomial of degree λ on the variables x..., x n X s λ (x) = T where T runs over all semistandard tableaux of shape λ on the alphabet {,..., n}. Example λ = (, ), λ = 3 n = 3 X α(t ) s λ (x, x, x 3 ) = x x + x x 3 + x x + x x x 3 + x x 3 + x x 3 + x x 3. 3 / 9

32 Kostka number K λ, α is the number of semistandard tableaux of shape λ and type α. The Schur function on the variables x,..., x n with α + + α n = λ. s n (λ, x) = X α Z n 0 K λ, α x α, 3 / 9

33 K λ β = K λ α, with β any permutation of α. 33 / 9

34 K λ β = K λ α, with β any permutation of α. Corollary The Schur function s(λ, x) = P α weak composition of λ homogeneous symmetric function in x,..., x n. K λ, α x α, is a 34 / 9

35 Product of Schur functions The Schur functions s λ form an additive basis for the ring of the symmetric functions. A product of Schur functions s µ s ν can be expressed as a non-negative integer linear sum of Schur functions: s µ s ν = X λ c λ µ νs λ. 35 / 9

36 What does c λ µν count? 36 / 9

37 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 37 / 9

38 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 38 / 9

39 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 39 / 9

40 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 40 / 9

41 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 4 / 9

42 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 4 / 9

43 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 43 / 9

44 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 44 / 9

45 . Littlewood-Richardson rule µ = (3, ), ν = (, ) 45 / 9

46 . Littlewood-Richardson rule µ = (3, ), ν = (, ) invalid tableaux 46 / 9

47 Littlewood-Richardson rule µ = (3, ), ν = (, ) 47 / 9

48 Littlewood-Richardson rule µ = (3, ), ν = (, ) 48 / 9

49 Littlewood-Richardson rule µ = (3, ), ν = (, ) 49 / 9

50 Littlewood-Richardson rule µ = (3, ), ν = (, ) 50 / 9

51 Littlewood-Richardson rule µ = (3, ), ν = (, ) 5 / 9

52 Littlewood-Richardson rule µ = (3, ), ν = (, ) 5 / 9

53 Littlewood-Richardson rule µ = (3, ), ν = (, ) 53 / 9

54 Littlewood-Richardson rule µ = (3, ), ν = (, ) s µ s ν = s 53 + s 5 + s 43 + s 4 + s 4 + s 33 + s 3 54 / 9

55 µ = (3, ), ν = (, ) 55 / 9

56 µ = (3, ), ν = (, ) 56 / 9

57 µ = (3, ), ν = (, ) 57 / 9

58 µ = (3, ), ν = (, ) 58 / 9

59 µ = (3, ), ν = (, ) 59 / 9

60 µ = (3, ), ν = (, ) 60 / 9

61 µ = (3, ), ν = (, ) 6 / 9

62 µ = (3, ), ν = (, ) 6 / 9

63 µ = (3, ), ν = (, ) 63 / 9

64 µ = (3, ), ν = (, ) 64 / 9

65 µ = (3, ), ν = (, ) 65 / 9

66 µ = (3, ), ν = (, ) s µ s ν = s 5 + s 5 + s 43 + s 4 + s 4 + s 33 + s 3 + s 3 66 / 9

67 µ = (3, ), ν = (, ) s µ s ν = s 5 + s 5 + s 43 + s 4 + s 4 + s 33 + s 3 + s 3 c 4 µ ν = 67 / 9

68 µ = (3, ), ν = (, ) s µ s ν = s 5 + s 5 + s 43 + s 4 + s 4 + s 33 + s 3 + s 3 c 4 µ ν = 68 / 9

69 Littlewood-Richardson rule c λ µν is the number of tableaux with shape λ/µ and content ν satisfying If one reads the labeled entries in reverse reading order, that is, from right to left across rows taken in turn from bottom to top, at any stage, the number of i s encountered is at least as large as the number of (i + ) s encountered, # s # s.... µ 3 3 λ ν = (5, 3, ) 69 / 9

70 3.Integer Hives (99) Knutson-Tao (99) An n-integer hive is a triangular graph made of ( n + ) + ( n ) = n unitary triangles and ( n + ) vertices with non-negative edge labels satisfying a set of conditions given by linear inequalities called hive conditions n = 5 70 / 9

71 (Edge) Hive conditions Two distinct types of elementary triangles with non-negative integer edge labelling ρ σ τ τ σ ρ σ + τ = ρ σ + τ = ρ 7 / 9

72 (Edge) Hive conditions continued Three distinct types of rhombi with non-negative integer edge labelling δ δ α α α γ β δ β γ β α γ β δ α + δ = β + γ γ 7 / 9

73 Hive conditions continued δ δ α α α γ β δ β γ β α γ β δ α + δ = β + γ γ 73 / 9

74 Hive boundary conditions overlapping pairs of rhombi: edge labels along any line parallel to north-west, north-east and southern boundaries are weakly decreasing in the north-east, south-east and easterly directions. µ ν γ λ β α α β β γ α γ α β γ 74 / 9

75 Knutson-Tao Hives 99 The Littlewood-Richardson coefficients c λ µ ν is the number of Hives with boundary µ, ν and λ. 75 / 9

76 4. Horn conjecture (6) 76 / 9

77 4. Horn conjecture (6) There exist n n Hermitian matrices A, B and C, with integer eigenvalues arranged in weakly decreasing order µ = (µ,..., µ n ), ν = (ν,..., ν n ) and λ = (λ,..., λ n ) respectively, such that C = A + B if and only if µ, ν and λ satisfy a certain huge system of linear inequalities. 77 / 9

78 Horn inequalities Let N = {,,..., n}, then for fixed d, with d n, let I = {i, i,..., i d } N. 78 / 9

79 Horn inequalities Let N = {,,..., n}, then for fixed d, with d n, let I = {i, i,..., i d } N. Let I, J, K N with #I = #J = #K = d and ordered decreasingly. One defines the partitions α(i ) = I (d,...,, ), β(j) = J (d,...,, ), γ(k) = K (d,...,, ). 79 / 9

80 Horn inequalities Let N = {,,..., n}, then for fixed d, with d n, let I = {i, i,..., i d } N. Let I, J, K N with #I = #J = #K = d and ordered decreasingly. One defines the partitions α(i ) = I (d,...,, ), β(j) = J (d,...,, ), γ(k) = K (d,...,, ). Let Td n be the set of all triples (I, J, K) with I, J, K N and #I = #J = #K = d such that c γ(k) α(i ),β(j) > / 9

81 Horn inequalities continued µ, ν, λ are said to satisfy the Horn inequalities if nx k= X for all triples (I, J, K) T n d λ k = nx i= k K λ k X i I µ i + nx ν j j= µ i + X j J ν j with d =,..., n. Not all of Horn s inequalities are essential. The essential inequalities are those for which (I, J, K) satisfy c γ(k) α(i ),β(j) =. 8 / 9

82 Horn inequalities and Littlewood-Richardson coefficients c λ µ,ν > 0 if and only if the Horn inequalities are satisfied nx k= X λ k = nx nx µ i + ν j i= j= X X λ k µ i + ν j k K i I j J for all triples (I, J, K) Td n with d =,..., n. 8 / 9

83 Where do Horn inequalities come from? Impose on a n-hive a puzzle of size n. 83 / 9

84 5. Knutson-Tao-Woodward Puzzles (04) A puzzle of size n is a tiling of an equilateral triangle of side length n with puzzle pieces each of unit side length. Puzzle pieces may be rotated in any orientation but not reflected, and wherever two pieces share an edge, the numbers on the edge must agree. 84 / 9

85 5. Knutson-Tao-Woodward Puzzles (04) A puzzle of size n is a tiling of an equilateral triangle of side length n with puzzle pieces each of unit side length. Puzzle pieces may be rotated in any orientation but not reflected, and wherever two pieces share an edge, the numbers on the edge must agree µ 0 ν λ 85 / 9

86 Partitions and 0-strings Fix positive integers 0 < d < n and consider a d (n d) rectangle. 86 / 9

87 Partitions and 0-strings Fix positive integers 0 < d < n and consider a d (n d) rectangle. d = 4 n = 0 d = n d = 6 λ 0 0 λ = (4,,, 0) λ = (6, 5, 4, ) / 9

88 Puzzle rule (Knutson-Tao-Woodward) c λ µ ν is the number of puzzles with µ, ν and λ appearing as 0-strings along the boundary µ 0 ν λ 88 / 9

89 Horn triples and puzzles (I, J, K) is Horn triple if it specifies the positions of the 0 s on the boundary of any puzzle. It is essential if the puzzle with these boundary 0 s edges is unique. 89 / 9

90 Example I = {, 3}, J = {, 4}, K = {, 4} is a Horn triple since I, J, and K specify the positions of the 0 s on the boundary of the puzzle 90 / 9

91 Example continued I = {, 3}, J = {, 4}, K = {, 4} is a Horn triple. Superimpose the puzzle, with pink edges specified by those sets I, J, K, on a hive of size 5 and explore the hive conditions µ 5 ν µ 4 µ 3 s t ν ν 3 µ µ r q x y z w u ν 4 ν 5 λ λ λ 3 λ 4 λ 5 λ + λ 4 = x + y + w + u µ + y + w + u µ + z + y + ν 4 µ + z + y + ν 4 = µ + q + ν 4 µ + r + ν 4 µ + µ 3 + s + ν 4 µ + µ 3 + ν + ν 4 9 / 9

92 Sources A. Horn, Pacific J. Math Robert C. Thompson, , Contemp. Math., 47, Amer. Math. Soc., Providence, RI, 985. O. Azenhas, E. Marques de Sá, Linear and Multilinear Algebra 7 (990), no. 4, 9 4. A. A. Klyachko, Selecta Math. (N.S.) 4 (998), no. 3, W. Fulton, Séminaire Bourbaki. Vol. 997/98. Astérisque No. 5 (998), Exp. No. 845, 5, A. Knutson, T. Tao, J. Amer. Math. Soc. (999) A. P. Santana, J. F. Queiró, and E. Marques de Sá, Linear and Multilinear Algebra 46 (999), no. -, 3 W. Fulton, Bull. Amer. Math. Soc. (N.S.) 37 (000), no. 3, A. S. Buch, Enseign. Math. () 46 (000), no. -, A. Knutson, T. Tao and C. Woodward, Amer. Math. Soc. 7 (004) 948 Ronald C. King, C. Tollu, F. Toumazet, J. Combin. Theory, Series A,6 (009), / 9