Kostka number

For any partition ${\displaystyle \lambda }$, the Kostka number ${\displaystyle K_{\lambda \lambda }}$ is equal to 1: the unique way to fill the Young diagram of shape ${\displaystyle \lambda =(\lambda _{1},\dotsc ,\lambda _{m})}$ with ${\displaystyle \lambda _{1}}$ copies of 1, ${\displaystyle \lambda _{2}}$ copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape ${\displaystyle \lambda }$.)

The Kostka number ${\displaystyle K_{\lambda \mu }}$ is positive (i.e., there exist semistandard Young tableaux of shape ${\displaystyle \lambda }$ and weight ${\displaystyle \mu }$) if and only if ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are both partitions of the same integer ${\displaystyle n}$ and ${\displaystyle \lambda }$ is larger than ${\displaystyle \mu }$ in dominance order.[2]

In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if ${\displaystyle \mu =(1,1,\dotsc ,1)}$ is the partition whose parts are all 1 then a semistandard Young tableau of weight ${\displaystyle \mu }$ is a standard Young tableau; the number of standard Young tableaux of a given shape ${\displaystyle \lambda }$ is given by the hook-length formula.

An important simple property of Kostka numbers is that ${\displaystyle K_{\lambda \mu }}$ does not depend on the order of entries of ${\displaystyle \mu }$. For example, ${\displaystyle K_{(3,2)(1,1,2,1)}=K_{(3,2)(1,1,1,2)}}$. This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape ${\displaystyle \lambda }$ and weights ${\displaystyle \mu }$ and ${\displaystyle \mu ^{\prime }}$, where ${\displaystyle \mu }$ and ${\displaystyle \mu ^{\prime }}$ differ only by swapping two entries.[3]

In addition to the purely combinatorial definition above, they can also be defined as the coefficients that arise when one expresses the Schur polynomial ${\displaystyle s_{\lambda }}$ as a linear combination of monomial symmetric functions ${\displaystyle m_{\mu }}$:

${\displaystyle s_{\lambda }=\sum _{\mu }K_{\lambda \mu }m_{\mu },}$

where ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are both partitions of ${\displaystyle n}$. Alternatively, Schur polynomials can also be expressed[4] as

${\displaystyle s_{\lambda }=\sum _{\alpha }K_{\lambda \alpha }x^{\alpha },}$

where the sum is over all weak compositions ${\displaystyle \alpha }$ of ${\displaystyle n}$ and ${\displaystyle x^{\alpha }}$ denotes the monomial ${\displaystyle x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\dotsc x_{n}^{\alpha _{n}}}$.

On the level of representations of the symmetric group ${\displaystyle S_{n}}$, Kostka numbers express the decomposition of the permutation module ${\displaystyle M_{\mu }}$ in terms of the irreducible representations ${\displaystyle V_{\lambda }}$ where ${\displaystyle \lambda }$ is a partition of ${\displaystyle n}$, i.e.,

${\displaystyle M_{\mu }=\bigoplus _{\lambda }K_{\lambda \mu }V_{\lambda }.}$

On the level of representations of the general linear group ${\displaystyle \mathrm {GL} _{d}(\mathbb {C} )}$, the Kostka number ${\displaystyle K_{\lambda \mu }}$ also counts the dimension of the weight space corresponding to ${\displaystyle \mu }$ in the unitary irreducible representation ${\displaystyle U_{\lambda }}$ (where we require ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ to have at most ${\displaystyle d}$ parts).

The Kostka numbers for partitions of size at most 3 are as follows:

${\displaystyle K_{\varnothing \varnothing }=1,}$

${\displaystyle K_{(1)(1)}=1,}$

${\displaystyle K_{(2)(2)}=K_{(2)(1,1)}=1,}$

${\displaystyle K_{(1,1)(2)}=0,\,K_{(1,1)(1,1)}=1,}$

${\displaystyle K_{(3)(3)}=K_{(3)(2,1)}=K_{(3)(1,1,1)}=1,}$

${\displaystyle K_{(2,1)(3)}=0,\,K_{(2,1)(2,1)}=1,\,K_{(2,1)(1,1,1)}=2,}$

${\displaystyle K_{(1,1,1)(3)}=K_{(1,1,1)(2,1)}=0,\,K_{(1,1,1)(1,1,1)}=1.}$

These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions:

${\displaystyle s_{\varnothing }=m_{\varnothing }=1}$

${\displaystyle s_{(1)}=m_{(1)}}$

${\displaystyle s_{(2)}=m_{(2)}+m_{(1,1)}}$

${\displaystyle s_{(1,1)}=m_{(1,1)}}$

${\displaystyle s_{(3)}=m_{(3)}+m_{(2,1)}+m_{(1,1,1)}}$

${\displaystyle s_{(2,1)}=m_{(2,1)}+2m_{(1,1,1)}}$

${\displaystyle s_{(1,1,1)}=m_{(1,1,1)}}$

Kostka (1882, pages 118-120) gave tables of these numbers for partitions of numbers up to 8.

Kostka numbers are special values of the 1 or 2 variable Kostka polynomials:

${\displaystyle K_{\lambda \mu }=K_{\lambda \mu }(1)=K_{\lambda \mu }(0,1).}$

• Stanley, Richard (1999), Enumerative combinatorics, volume 2, Cambridge University Press
• Kostka, C. (1882), "Über den Zusammenhang zwischen einigen Formen von symmetrischen Funktionen", Crelle's Journal, 93: 89–123
• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144, archived from the original on 2012-12-11
• Sagan, Bruce E. (2001) [1994], "Schur functions in algebraic combinatorics", Encyclopedia of Mathematics, EMS Press