The Computation of Kostka Numbers and LittlewoodRichardson coefficients

The Computation of Kostka Numbers and Littlewood-Richardson coefficients is #P-complete Hariharan Narayanan University of Chicago

Young tableaux 1 1 1 2 2 2 3 4 3 A Young tableau of shape λ = (4, 3, 2) and content μ = (3, 3, 2, 1). §The numbers in each row are non-decreasing from the left to the right. §The numbers in each column are strictly increasing from the top to the bottom.

Skew tableaux A skew tableau of shape (2)*(2, 1) and content μ = (1, 1, 2, 1). 1 2 3 4 3 §As with tableaux, the numbers in each row of a skew tableau are non-decreasing from the left to the right, and the numbers in each column are strictly increasing from the top to the bottom.

LR (skew) tableaux §A (skew) tableau is said to be LR if, when its entries are read right to left, top to bottom, at any moment, the number of copies of i encountered is not less than the number of copies of i+1 encountered, for each i. 1 1 1 3 2 2 3 An LR skew tableau of shape (2)*(2, 1) and content (2, 2, 1). 1 2 2 A skew tableau of shape (2)*(2, 1) and content (2, 2, 1) that is not LR.

Kostka numbers §The Kostka number Kλμ is the number of Young tableaux having shape λ and content μ. If λ = (4, 3, 2) and μ = (3, 3, 2, 1), Kλμ =4 and the tableaux are - 1 1 1 2 2 4 3 3 2 1 1 1 2 2 2 3 4 3 1 1 1 2 2 3 3 4 1 1 1 2 2 2 3 3 2 4

Littlewood-Richardson Coefficients Let α and λ be partitions and ν be a vector with non-negative integer components. The Littlewood-Richardson coefficient cλα ν is the number of LR skew tableaux of shape λ*α that have content ν. If λ = (2, 1), α = (2, 1) and ν = (3, 2, 1), cλα ν=2 and the LR skew tableaux are 1 1 1 2 2 1 2 3 1 3 2 1

Representation theory Consider the group SLn(C) of n×n matrices over complex numbers that have determinant 1. Any matrix G = (gij) in SL(n, C) can be defined to act upon the formal variable xi by xi Σ gij xj. This leads to an action on the vectorspace T of all polynomials f in the variables {xi} according to G(f) x = f G-1 x.

Representation theory One can decompose T into the direct sum of vectorspaces Vλ, where λ = (λ 1 , …, λk) ranges over all partitions (of all natural numbers n) such that each Vλ is invariant under the action of SLn(C) and cannot be decomposed further into the non-trivial sum of SLn(C) invariant subspaces.

Representation theory Let H be the subgroup of SLn(C) consisting of all diagonal matrices. Though Vλ cannot be split into the direct sum of SLn(C) - invariant subspaces, it can be split into the direct sum of one dimensional H-invariant subspaces Vλμ Vλ = + Vλμ +K λμ , μ where μ ranges over all partitions of the same number n that λ is a partition of, and the multiplicity with which Vλμ occurs in Vλ is K λμ , the Kostka number defined earlier.

Representation theory The Littlewood-Richardson coefficient appears as the multiplicity of Vν in the decomposition of the tensor product Vλ◙Vα : Vλ◙Vα = + Vν +cλα ν The Kostka numbers and the Littlewood Richardson coefficients also play an essential role in the representation theory of the symmetric groups [F 97].

Related work § Probabilistic polynomial time algorithms exist, that calculate the set of all non-zero Kostka numbers, and Littlewood. Richardson coefficients, for certain fixed parameters, in time, polynomial in the total size of the input and output [BF 97]. § Vector partition functions have been used to calculate Kostka numbers and Littlewood-Richardson coefficients [C 03], and to study their properties [BGR 04]. § In his thesis, E. Rassart described some polynomiality properties of Kostka numbers and Littlewood-Richardson coefficients, and asked if they could be computed in polynomial time [R 04].

The problems are in #P Kostka numbers : § A tableau is fully described by the number of copies of j that are present in its ith row. This description has polynomial length. § Given such a description, one can verify whether it corresponds to a tableau of the required kind, in polynomial time, by checking § whether the columns have strictly increasing entries, from the top to the bottom, and that § the shape and content are correct.

The problems are in #P Littlewood Richardson coefficients : § An LR skew tableau λ*α is fully described by its shape and the number of copies of j that are present in the ith row of it. § This description has polynomial length. § Given such a description, one can verify whether it corresponds to an LR skew tableau of the required kind by checking § whether the columns have strictly increasing entries, from the top to the bottom, § that the shape and content are correct and § that the skew tableau is LR. § Each can be verified in polynomial time.

Hardness Results n We prove that the computation of the Kostka number Kλμ is #P hard, by reducing to it, the #P complete problem [DKM 79] of computing the number of 2 × k contingency tables with given row and column sums. n The problem of computating the Littlewood – Richardson coefficient cλα ν is shown to be #P-hard by reducing to it, the computation of the Kostka number Kλμ.

Contingency tables n A contingency table is a matrix of non-negative integers having prescribed row and column sums. 4 3 3 2 2

Contingency tables Counting the number of 2 × k contingency tables with given row and column sums is #P complete. [DKM 79] Example: A contingency table with row sums a = (4, 3) and column sums b = (3, 2, 2) : - 2 1 1 4 1 1 1 3 3 2 2

Reduction to computing Kostka numbers We shall exhibit a reduction from the problem of counting the number of 2 × k contingency tables with row sums a = (a 1, a 2), a 1 ≥ a 2 and column sums b = (b 1, …, bk) to the set of Young tableau having shape n λ = (a 1+a 2, a 2) and content μ = (b, a 2).

RSK correspondence n The Robinson Schensted Knuth (RSK) correspondence gives a bijection between the set I(a, b) of contingency tables having row sums a, column sums b, and the set U T(λ’, a) × T(λ’, b) of pairs of tableaux having contents a and b respectively.

C RSK correspondence 2 1 1 1 P Q

RSK correspondence 1 1 1 P 1 Q 1

RSK correspondence 0 1 1 1 P 1 Q 1 1 1

RSK correspondence 0 0 1 1 P 1 Q 1 2 1 1 1

RSK correspondence 0 0 0 1 1 1 P 1 Q 1 2 3 1 1

RSK correspondence 0 0 1 1 P Q 1 1 1 2 3 1 1

RSK correspondence 0 0 1 1 P 1 2 Q 1 1 3 1 2 1 1 1

RSK correspondence 0 0 0 1 P Q 2 1 1 3 1 2 1 1 1

RSK correspondence 0 0 0 1 P Q 1 1 2 3 1 2 1 1 2 2 1 1

RSK correspondence 0 0 0 Content P = (3, 2, 2) = b Content Q = (4, 3) = a P Q 1 1 2 3 1 1 2 2 1 1 2

RSK correspondence n Thus, the RSK correspondence gives us the identity |I(a, b)| = Σ Kλ’a. Kλ’b § Kλ’a > 0 implies that λ 1 ≥ a 1 and that λ 2 ≤a 2 , but b is arbitrary, so the summation is over a set exponential in the size of (a, b).

RSK correspondence §Q is fully determined by its shape and content since it has only 1’s and 2’s. Content P = b Content Q = a In other words Kλ’a > 0 implies that Kλ’a = 1. P 1 1 2 3 1 Q 2 3 1 1 2 2 1 1 2

RSK correspondence §The shape of P and Q could be any s = (s_1, s_2) such that s_1 ≥ a_1 and s_2 ≤a_2, but none other. Content P = b Content Q = a P 1 1 2 3 1 Q 2 3 1 1 2 2 1 1 2

Reduction to computing Kostka numbers §Extend P by padding it with copies of k+1 to a tableau T of shape λ and content μ, where λ = (a_1 + a_2, a_2) and μ = (b, a_2). 1 1 1 2 3 4 4 In our example, shape λ = (7, 3), and content μ = (3, 2, 2, 3).

From Contingency tables to Kostka numbers 2 1 1 1 1 2 3 4 4 Row sums a = content of Q= (4, 3), Column sums b = content of P = (3, 2, 2), shape λ = (7, 3), and content μ = (3, 2, 2, 3). 1 1 2 3 1 1 2 2 1 2 3 P 1 1 2 Q

From Kostka numbers to Littlewood -Richardson coefficients Given a shape λ, content μ = (μ 1, …, μs ), let α = (μ 2, μ 2+μ 3 , …, μ 2+…+μs), and let ν = λ + α Claim: Kλμ = cλα ν

From Kostka numbers to Littlewood -Richardson coefficients T S 1 1 1 2 3 4 2 1 1 1 2 3 4 3 2 4 3 4 4 4 1 1 1 2 2 2 3 3 3 1 1

From Kostka numbers to Littlewood -Richardson coefficients Any tableau of shape λ and content μ, can be embedded in an LR skew tableau of shape λ*α, and content ν. n 1 1 1 2 3 4 4

From Kostka numbers to Littlewood -Richardson coefficients Any tableau of shape λ and content μ, can be embedded in an LR skew tableau of shape λ*α, and content ν. n 1 1 1 2 3 4 4 1 1 1 2 2 2 3 3 3 1 1

From Kostka numbers to Littlewood -Richardson coefficients Any LR skew tableau of shape λ*α and content ν, when restricted to λ, has content μ. n 1 1 1 2 3 4 4 1 1 1 2 2 2 3 3 3 1 1

From Kostka numbers to Littlewood -Richardson coefficients Any LR skew tableau of shape λ*α and content ν, when restricted to λ, has content μ. n 1 1 1 2 3 4 4

Future directions n Do there exist Fully Polynomial Randomized Approximation Schemes (FPRAS) for the evaluation of these quantities?

Thank You!

References n n n § § § [BF 97] A. Barvinok and S. V. Fomin, Sparse interpolation of symmetric polynomials, Advances in Applied Mathematics, 18 (1997), 271 -285, MR 98 i: 05164. [BGR 04] S. Billey, V. Guillimin, E. Rassart, A vector partition function for the multiplicities of slk(C), Journal of Algebra, 278 (2004) no. 1, 251 -293. [C 03] C. Cochet, Kostka numbers and Littlewood-Richardson coefficients, preprint (2003). [DKM 79] M. Dyer, R. Kannan and J. Mount, Sampling Contingency tables, Random Structures and Algorithms, (1979) 10 487 -506. [F 97] W. Fulton, Young Tableaux, London Mathematical Society Student Texts 35 (1997). [R 04] E. Rassart, Geometric approaches to computing Kostka numbers and Littlewood-Richardson coefficients, preprint