The Computation of Kostka The Computation of Kostka Numbers and Littlewood-Numbers and Littlewood-Richardson coefficients is Richardson coefficients is

#P-complete#P-completeHariharan NarayananHariharan Narayanan

University of ChicagoUniversity of Chicago

Young tableaux Young tableaux

1 1 1

2

3

3

2 2

4A 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 Skew tableaux

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.

1 3

3

2

4

A skew tableau of shape (2)*(2,1) and content μ = (1, 1, 2, 1).

LR (skew) tableaux 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 encounteredis not less than the number of copies of i+1 encountered, for each i. 1 1

2

2

3

An LR skew tableau of shape (2)*(2,1) and content (2, 2, 1).

1 1

2

3

2

A skew tableau of shape (2)*(2,1) and content (2, 2, 1) that is not LR.

Kostka numbersKostka numbers

The Kostka number Kλμ is the number of Young tableaux having shape λ and content μ.

1 1 1

2

2

3

2 3

4

1 1 1

2

2

3

2 4

3

1 1 1

2

4

3

2 2

3

1 1 1

2

3

3

2 2

4

If λ = (4, 3, 2) and μ = (3, 3, 2, 1), Kλμ =4 and the tableaux are -

Littlewood-Richardson Littlewood-Richardson CoefficientsCoefficients

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.

1 1

1

2

3

1 1

1

2

2

If λ = (2, 1), α = (2, 1) and ν = (3, 2, 1), cλα ν=2 and the LR skew tableaux are -

2 3

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-1x.

Representation theoryRepresentation theory

One can decompose T into the direct sum of vectorspaces Vλ, where λ = (λ1 , …, 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 theoryRepresentation 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 ,

μwhere 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 theoryRepresentation theory

The Littlewood-Richardson coefficient appears as the multiplicity of Vν in the decomposition of the tensor productVλ◙Vα :-

Vλ◙Vα = + Vν +cλα

ν

The Kostka numbers and the Littlewood Richardson coefficients also play an essential role in the representationtheory of the symmetric groups [F97].

Representation theoryRepresentation theory

Related workRelated work Probabilistic polynomial time algorithms exist, that Probabilistic polynomial time algorithms exist, that

calculate the set of all non-zero Kostka numbers, and calculate the set of all non-zero Kostka numbers, and Littlewood-Richardson coefficients, for certain fixed Littlewood-Richardson coefficients, for certain fixed parameters, in time, polynomial in the total size of the parameters, in time, polynomial in the total size of the input and output [BF97].input and output [BF97].

Vector partition functions have been used to calculate Vector partition functions have been used to calculate Kostka numbers and Littlewood-Richardson coefficients Kostka numbers and Littlewood-Richardson coefficients

[C03], and to study their properties [BGR04].[C03], and to study their properties [BGR04].

In his thesis, E. Rassart described some polynomiality In his thesis, E. Rassart described some polynomiality properties of Kostka numbers and Littlewood-properties of Kostka numbers and Littlewood-Richardson coefficients, and asked if they could be Richardson coefficients, and asked if they could be computed in polynomial time [R04].computed in polynomial time [R04].

The problems are in #PThe problems are in #P

Kostka numbers :Kostka numbers : A tableau is fully described by the number of A tableau is fully described by the number of

copies ofcopies of j j that are present in its ithat are present in its ith th row. This row. This description has polynomial length.description has polynomial length.

Given such a description, one can verify whether Given such a description, one can verify whether it corresponds to a tableau of the required kind, it corresponds to a tableau of the required kind, in polynomial time, by checkingin polynomial time, by checking whether the columns have strictly increasing whether the columns have strictly increasing

entries, from the top to the bottom, and that entries, from the top to the bottom, and that the shape and content are correct.the shape and content are correct.

The problems are in #PThe problems are in #P

Littlewood Richardson coefficients :Littlewood Richardson coefficients : An LR skew tableau An LR skew tableau λ*α is fully described by its is fully described by its

shape and the number of copies of shape and the number of copies of jj that are that are present in the ipresent in the ithth row of row of it. .

This description has polynomial length. This description has polynomial length. Given such a description, one can verify whether Given such a description, one can verify whether

it corresponds to an LR skew tableau of the it corresponds to an LR skew tableau of the required kind by checkingrequired kind by checking whether the columns have strictly increasing whether the columns have strictly increasing

entries, from the top to the bottom, entries, from the top to the bottom, that the shape and content are correct andthat the shape and content are correct and that the skew tableau is LR.that the skew tableau is LR.

Each can be verified in polynomial time.Each can be verified in polynomial time.

Hardness ResultsHardness Results We prove that the computation of the Kostka We prove that the computation of the Kostka

number number Kλμ is #P hard, by reducing to it, the #P is #P hard, by reducing to it, the #P complete problem [DKM79] of computing the complete problem [DKM79] of computing the number of 2 × k contingency tables with given number of 2 × k contingency tables with given row and column sums. row and column sums.

The problem of computating the Littlewood – The problem of computating the Littlewood – Richardson coefficient Richardson coefficient cλα ν is shown to be #P-is shown to be #P-hard by reducing to it, the computation of the hard by reducing to it, the computation of the Kostka number Kostka number Kλμ..

Contingency tablesContingency tables A contingency table is a matrix of non-A contingency table is a matrix of non-

negative integers having prescribed row negative integers having prescribed row and column sums.and column sums.

44

33

33 22 22

Contingency tablesContingency tables

Counting the number of 2 × k contingency tables with given Counting the number of 2 × k contingency tables with given row and column sums is #P complete. row and column sums is #P complete. [DKM79][DKM79]

Example:Example:A contingency table with row sums a = (4, 3) and column sums A contingency table with row sums a = (4, 3) and column sums

b = (3, 2, 2) :-b = (3, 2, 2) :-

2 2 11 11

11 11 11

44

33

33 22 22

We shall exhibit a reduction from the problem We shall exhibit a reduction from the problem

of counting the number of 2 × k contingency of counting the number of 2 × k contingency

tables with row sums a = (atables with row sums a = (a11, a, a22), a), a11 ≥ a ≥ a22

and column sums b = (band column sums b = (b11, …, b, …, bkk))

to the set of Young tableau having shape to the set of Young tableau having shape

λλ = (a = (a11+a+a22, a, a22) and content ) and content μμ = (b, a = (b, a22).).

Reduction to computing Kostka numbers

The Robinson Schensted Knuth (RSK) The Robinson Schensted Knuth (RSK) correspondence gives a bijection between the set correspondence gives a bijection between the set I(a, b) of contingency tables having row sums a, I(a, b) of contingency tables having row sums a, column sums b, column sums b,

and the set and the set UU T( T(λλ’, a) × T(’, a) × T(λλ’, b) of pairs of ’, b) of pairs of

tableaux having contents a and b respectively.tableaux having contents a and b respectively.

RSK correspondenceRSK correspondence

RSK correspondenceRSK correspondence

2 2 11 11

11 11 11

P Q

C

RSK correspondenceRSK correspondence

1 1 11 11

11 11 11

P Q

11

RSK correspondenceRSK correspondence

00 11 11

11 11 11

P Q

11 1 1

RSK correspondenceRSK correspondence

00 00 11

11 11 11

P Q

11 1 12 1

RSK correspondenceRSK correspondence

00 00 00

11 11 11

P Q

11 1 12 1 13

RSK correspondenceRSK correspondence

00 00 00

00 11 11

P Q

11 1 12 1 13

1

RSK correspondenceRSK correspondence

00 00 00

00 11 11

P Q

11 1 1

2

1 131

2

RSK correspondenceRSK correspondence

00 00 00

00 00 11

P Q

11 1 1

2

1 131

2

2

RSK correspondenceRSK correspondence

00 00 00

00 00 11

P Q

11 1 1

2

1 11

2

2

3 2

RSK correspondenceRSK correspondence

00 00 00

00 00 00

P Q

11 1 1

2

1 11

2

2

3 2

3 2

Content P = (3, 2, 2) = bContent Q = (4, 3) = a

RSK correspondenceRSK correspondence

Thus, the RSK correspondence gives us the identity |I(a, b)| Thus, the RSK correspondence gives us the identity |I(a, b)| = = ΣΣ K Kλλ’a’aKKλλ’b’b

KKλλ’a ’a > 0> 0 implies thatimplies that λ1 ≥ a1 and that λ2 ≤a2 , but b is arbitrary, so the summation is over a set exponential in the size of (a, b).

RSK correspondenceRSK correspondence

P Q

11 1 1

2

1 11

2

2

3 2

3 2

Content P = bContent Q = a

Q is fully determined by its shape and content since it has only 1’s and 2’s.

In other words KKλλ’a ’a > 0> 0

implies that Kimplies that Kλλ’a ’a = 1.= 1.

RSK correspondenceRSK correspondence

P Q

11 1 1

2

1 11

2

2

3 2

3 2

Content P = bContent Q = a

The shape of P and Qcould be any s = (s_1, s_2)such that s_1 ≥ a_1 and s_2 ≤a_2,but none other.

In our example, shape λ = (7, 3), and content μ = (3, 2, 2, 3).

1 1

2

1 2

3

3 4 4

4

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).

Reduction to computing Kostka numbers

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

1 2

3

3 4 4

4

2 2 11 11

11 11 11

1 1

2

1 2

3

3

1 1 1 1

2 2

2

From Contingency tables to Kostka numbers

P

Q

From Kostka numbers to From Kostka numbers to Littlewood-Richardson Littlewood-Richardson

coefficientscoefficientsGiven a shape λ, content μ = (μ1,…, μs ), let α = (μ2, μ2+μ3 , …, μ2+…+μs), and let ν = λ + α

Claim: Kλμ = cλα ν

From Kostka numbers to From Kostka numbers to Littlewood-Richardson Littlewood-Richardson

coefficientscoefficients

1 1

2

1 2

3

3 4 4

4

1 1

2

1 2

3

3 4 4

4

3

111111

22222

33

1

T

S

Any tableau of shape λ and content μ, can be

embedded in an LR skew tableau of shape

λ*α, and content ν.

1 1

2

1 2

3

3 4 4

4

From Kostka numbers to From Kostka numbers to Littlewood-Richardson Littlewood-Richardson

coefficientscoefficients

Any tableau of shape λ and content μ, can be

embedded in an LR skew tableau of shape

λ*α, and content ν.

1 1

2

1 2

3

3 4 4

4

3

111111

22222

33

1

From Kostka numbers to From Kostka numbers to Littlewood-Richardson Littlewood-Richardson

coefficientscoefficients

Any LR skew tableau of shape Any LR skew tableau of shape λλ**αα and and

content content νν, when restricted to , when restricted to λλ, has , has content content μ.

1 1

2

1 2

3

3 4 4

4

3

111111

22222

33

1

From Kostka numbers to From Kostka numbers to Littlewood-Richardson Littlewood-Richardson

coefficientscoefficients

Any LR skew tableau of shape Any LR skew tableau of shape λλ**αα and and

content content νν, when restricted to , when restricted to λλ, has , has content content μ.

1 1

2

1 2

3

3 4 4

4

From Kostka numbers to From Kostka numbers to Littlewood-Richardson Littlewood-Richardson

coefficientscoefficients

Future directionsFuture directions

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

Thank You!Thank You!

ReferencesReferences [BF97] A. Barvinok and S.V. Fomin, [BF97] A. Barvinok and S.V. Fomin, Sparse interpolation of Sparse interpolation of

symmetric polynomialssymmetric polynomials, Advances in Applied Mathematics, 18 , Advances in Applied Mathematics, 18 (1997), 271-285, MR 98i:05164.(1997), 271-285, MR 98i:05164.

[BGR04] S. Billey, V. Guillimin, E. Rassart, [BGR04] S. Billey, V. Guillimin, E. Rassart, A vector partition A vector partition function for the multiplicities of slfunction for the multiplicities of slkk(C), (C), Journal of Algebra, 278 Journal of Algebra, 278 (2004) no. 1, 251-293.(2004) no. 1, 251-293.

[C03] C. Cochet, Kostka numbers and Littlewood-Richardson [C03] C. Cochet, Kostka numbers and Littlewood-Richardson coefficients, preprint (2003).coefficients, preprint (2003).

[DKM79] M. Dyer, R. Kannan and J. Mount, [DKM79] M. Dyer, R. Kannan and J. Mount, Sampling Contingency Sampling Contingency tablestables, Random Structures and Algorithms, (1979) 10 487-506., Random Structures and Algorithms, (1979) 10 487-506.

[F97] W. Fulton, Young Tableaux, London Mathematical Society [F97] W. Fulton, Young Tableaux, London Mathematical Society Student Texts 35 (1997).Student Texts 35 (1997).

[R04] E. Rassart, Geometric approaches to computing Kostka [R04] E. Rassart, Geometric approaches to computing Kostka numbers and Littlewood-Richardson coefficients, preprint (2003).numbers and Littlewood-Richardson coefficients, preprint (2003).