The Computation of Kostka Numbers and LittlewoodRichardson coefficients is

1
The Computation of Kostka Numbers and
Littlewood-Richardson coefficients is P-complete

• Hariharan Narayanan
• University of Chicago

2
Young tableaux
1
1
1
3

• A Young tableau of shape ? (4, 3, 2) and
  content µ (3, 3, 2, 1).

2
2
2
3
4

• 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.

3
Skew tableaux

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

• 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
2
3
4
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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 i1
  encountered, for each i.

1
1
1
1
3
2
2
2
2
3

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

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

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Kostka numbers

• The Kostka number K?µ is the number of Young
  tableaux having shape ? and content µ.

1
1
1
2
If ? (4, 3, 2) and µ (3, 3, 2, 1), K?µ 4
and the tableaux are -
2
2
3
3
4
1
1
1
2
1
1
1
4
1
1
1
3
2
2
4
2
2
2
2
2
2
3
3
3
3
3
4
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Littlewood-Richardson Coefficients

• Let a and ? be partitions and ? be a vector with
  non-negative integer components.
• The Littlewood-Richardson coefficient c?a ? is
  the number of LR skew tableaux of shape ?a that
  have content ?.

If ? (2, 1), a (2, 1) and ? (3, 2, 1), c?a
?2 and the LR skew tableaux are -
1
1
1
1
2
2
1
2
1
3
3
2
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Representation theory

Consider the group SLn(C) of nn 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
S 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.
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Representation theory

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

Let H be the subgroup of SLn(C) consisting of
all diagonal matrices. Though V? 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 V?µ V?
V?µ K ?µ , µ where µ ranges over all
partitions of the same number n that ? is a
partition of, and the multiplicity with which V?µ
occurs in V? is K ?µ , the Kostka number
defined earlier.
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Representation theory

The Littlewood-Richardson coefficient appears as
the multiplicity of V? in the decomposition of
the tensor product V??Va - V??Va V? c?a
? The Kostka numbers and the Littlewood
Richardson coefficients also play an essential
role in the representation theory of the
symmetric groups F97.
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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 BF97.
• Vector partition functions have been used to
  calculate
• Kostka numbers and Littlewood-Richardson
  coefficients C03, and to study their properties
  BGR04.
• 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 R04.

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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.

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The problems are in P

• Littlewood Richardson coefficients
• An LR skew tableau ?a 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.

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Hardness Results

• We prove that the computation of the Kostka
  number K?µ is P hard, by reducing to it, the P
  complete problem DKM79 of computing the number
  of 2 k contingency tables with given row and
  column sums.
• The problem of computating the Littlewood
  Richardson coefficient c?a ? is shown to be
  P-hard by reducing to it, the computation of
  the Kostka number K?µ.

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Contingency tables

• A contingency table is a matrix of non-negative
  integers having prescribed row and column sums.

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Contingency tables

• Counting the number of 2 k contingency tables
  with given row and column sums is P complete.
  DKM79
• Example
• A contingency table with row sums a (4, 3) and
  column sums b (3, 2, 2) -

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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 (a1, a2), a1 a2
• and column sums b (b1, , bk)
• to the set of Young tableau having shape
• ? (a1a2, a2) and content µ (b, a2).
  

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RSK correspondence

• 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(?, a) T(?, b) of pairs
  of tableaux having contents a and b respectively.
  

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RSK correspondence
C
P Q
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RSK correspondence
P Q
1
1
21
RSK correspondence
P Q
1
1
1
1
22
RSK correspondence
P Q
1
1
1
1
2
1
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RSK correspondence
P Q
1
1
1
1
2
1
1
3
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RSK correspondence
P Q
1
1
1
1
1
2
1
1
3
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RSK correspondence
P Q
1
1
1
1
1
1
3
1
2
2
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RSK correspondence
P Q
2
1
1
1
1
1
1
3
1
2
2
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RSK correspondence
P Q
1
1
1
1
1
1
1
2
2
2
3
2
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RSK correspondence
Content P (3, 2, 2) b Content Q (4, 3) a
P Q
1
1
1
1
1
1
1
2
3
2
2
2
3
2
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RSK correspondence

• Thus, the RSK correspondence gives us the
  identity I(a, b) S K?aK?b
• K?a gt 0 implies that ?1 a1 and that ?2 a2
  , but b is arbitrary, so the summation is over a
  set exponential in the size of (a, b).

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RSK correspondence
Content P b Content Q a

• Q is fully determined by its
• shape and content since it has
• only 1s and 2s.
• In other words K?a gt 0
• implies that K?a 1.

P Q
1
1
1
1
1
1
1
2
3
2
2
2
3
2
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RSK correspondence
Content P b Content Q a

• 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.

P Q
1
1
1
1
1
1
1
2
3
2
2
2
3
2
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Reduction to computing Kostka numbers

• Extend P by padding it with copies of k1 to a
  tableau T of shape ? and content µ, where ?
  (a_1 a_2, a_2) and µ (b, a_2).

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


1
1
1
2
3
4
4
2
3
4
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From Contingency tables to Kostka numbers
1
1
1
2
3

P
2
3
Q
1
1
1
1
2
2
2
1
1
1
2
3
4
4
2
3
4
Row sums a content of Q (4, 3), Column sums b
content of P (3, 2, 2), shape ? (7, 3),
and content µ (3, 2, 2, 3).


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From Kostka numbers to Littlewood-Richardson
coefficients
Given a shape ?, content µ (µ1,, µs ), let a
(µ2, µ2µ3 , , µ2µs), and let ? ?
a Claim K?µ c?a ?
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From Kostka numbers to Littlewood-Richardson
coefficients
1
1
1
2
3
4
4
T
2
3
4
1
1
1
1
1
1
1
2
2
2
2
2
3
3
3
1
1
1
2
3
4
4
S
2
3
4
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From Kostka numbers to Littlewood-Richardson
coefficients

• Any tableau of shape ? and content µ, can be
• embedded in an LR skew tableau of shape
• ?a, and content ?.

1
1
1
2
3
4
4
2
3
4
37
From Kostka numbers to Littlewood-Richardson
coefficients

• Any tableau of shape ? and content µ, can be
• embedded in an LR skew tableau of shape
• ?a, and content ?.

1
1
1
1
1
1
1
2
2
2
2
2
3
3
3
1
1
1
2
3
4
4
2
3
4
38
From Kostka numbers to Littlewood-Richardson
coefficients

• Any LR skew tableau of shape ?a and
• content ?, when restricted to ?, has content µ.

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

• Any LR skew tableau of shape ?a and
• content ?, when restricted to ?, has content µ.

1
1
1
2
3
4
4
2
3
4
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Future directions

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

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• Thank You!

42
References

• BF97 A. Barvinok and S.V. Fomin, Sparse
  interpolation of symmetric polynomials, Advances
  in Applied Mathematics, 18 (1997), 271-285, MR
  98i05164.
• BGR04 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.
• C03 C. Cochet, Kostka numbers and
  Littlewood-Richardson coefficients, preprint
  (2003).
• DKM79 M. Dyer, R. Kannan and J. Mount, Sampling
  Contingency tables, Random Structures and
  Algorithms, (1979) 10 487-506.
• F97 W. Fulton, Young Tableaux, London
  Mathematical Society Student Texts 35 (1997).
• R04 E. Rassart, Geometric approaches to
  computing Kostka numbers and Littlewood-Richardson
  coefficients, preprint (2003).