Schur function




Representation theory



The Schur polynomials s λs_\lambda, are polynomials in nn variables indexed by partitions λ=λ 1λ 2λ n\lambda = \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n of nn, which constitute a linear basis of the ring of symmetric polynomials in nn variables


Given the partition λ 1λ 2λ n\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_n, the corresponding Schur polynomial is defined as follows. First define the n×nn\times n-determinant (for any partition α\alpha in nn parts)

a αdet(x i α j). a_\alpha \;\coloneqq\; det\big(x_i^{\alpha_j}\big).

Let δ=(n1,n2,,1,0)\delta=(n-1, n-2, \dots, 1, 0). Then the Schur polynomial attached to λ\lambda is the quotient

s λ(x 1,x 2,,x n)=a λ+δ/a δ. s_{\lambda}(x_1,x_2,\dots,x_n) = a_{\lambda+\delta}/a_{\delta}.

As is usual in the theory of symmetric functions one can also deal with formal power series in infinitely many variables. To make this precise one uses an inverse limit (see Macdonald 95) and obtains a Schur function s λs_{\lambda} for each partition, depending on countably many variables x 1,x 2,,x n,x n+1,x_1, x_2,\dots,x_n,x_{n+1}, \dots.


In terms of semistandard Young tableaux

A semistandard Young tableau is a Young tableau such that its values are:

  1. weakly increasing to the right (along rows),

  2. strictly increasing downwards (along columns).

For example:

1 1 1 3 2 3 4 \array{ 1 & 1 & 1 & 3 \\ 2 & 3 \\ 4 }

Given a semistandard Young tableau TT, we write X TX^T for the monomial which contains one factor of the variable x kx_k for each occurrence of kk in the Young tableau:

(1)x Tx 1 #1sx 1 #2s. x^T \;\coloneqq\; x_1^{\#1s} x_1^{\#2s} \cdots \,.

We write

(2)ssYT λssYY λ(N) ssYT_\lambda \;\supset\; ssYY_\lambda(N)

for the set of semistandard Young tableaux whose underlying partition (i.e. forgetting its labels) is λ\lambda, and, respectively, for its subset on those whose labels are bounded as T i,jNT_{i,j} \leq N.


For nn \in \mathbb{N} and λ\lambda a partition of nn, the corresponding Schur polynomial s λs_\lambda is equal to the sum over the monomials (1) associated with all semistandard Young tableaux (2) of shape λ\lambda:

s λ=TssYT(λ),x T. s_\lambda \;\;=\;\; \underset{ {T \in ssYT(\lambda)}, }{\sum} x^T \,.

(Sagan 01, Def. 4.4.1, review in Sagan Enc., p. 1)

This means that the Schur polynomial s λs_\lambda in NN variables is the sum over semistandard Young tableaux (2) of shape λ\lambda and with labels bounded as T i,jNT_{i,j} \leq N:

(3)s λ(x 1,,x N)=TssYT λ(N),x T. s_\lambda \big( x_1, \cdots, x_N \big) \;\;=\;\; \underset{ {T \in ssYT_\lambda(N)}, }{\sum} x^T \,.

In particular, the evaluation of a Schur polynomial at NN unit values is the number of semistandard Young tableaux with labels N\leq N:

(4)s λ(x 1=1,,x N=1)=TssYT λ(N),1=|ssYT λ(N)|. s_\lambda \big( x_1\!=\!1, \cdots, x_N \!=\! 1 \big) \;\;=\;\; \underset{ {T \in ssYT_\lambda(N)}, }{\sum} 1 \;\;=\;\; \left\vert ssYT_\lambda(N) \right\vert \,.

An immediate consequence is:


The coefficient of any monomial x 1 n 1x 2 n 2x_1^{n_1} x_2^{n_2} \cdots appearing in a Schur polynomial is non-negative.

In terms of Sym(n)Sym(n)-Characters

We discuss an expression of the Schur polynomials as symmetric polynomials with coefficients in the character-values of the symmetric group; this is Frobenius’ formula in Prop. below.

First to set up some notation:

For the present purpose, a partition of a natural number nn \in \mathbb{N} is a weakly decreasing sequence

λ=(λ 1λ 2λ k) \lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)

of natural numbers whose sum equals nn:

λ 1+λ 2++λ k=n. \lambda_1 + \lambda_2 + \cdots + \lambda_k \;=\; n \,.

By the representation theory of the symmetric group, such partitions label its irreducible representations in the form of Specht modules Specht (λ)Specht^{(\lambda)}. We write

(5)χ (λ)()Tr Specht (λ)() \chi^{(\lambda)}(-) \;\coloneqq\; Tr_{Specht^{(\lambda)}}(-)

for the irreducible character corresponding to this Specht module.

We write

Sym(n) CycLengths Part(n) Sym(n)/ adSym(n) \array{ Sym(n) & \overset{CycLengths}{\longrightarrow} & Part(n) \\ \big\downarrow & \nearrow_{\mathrlap{\simeq}} \\ Sym(n)/_{ad}Sym(n) }

for the function that sends a permutation σ\sigma of nn elements to the partition

(6)CycLengths(σ)=(l 1l 2l |Cycles(σ)|) CycLengths(\sigma) \;=\; (l_1 \geq l_2 \geq \cdots \geq l_{\left\vert Cycles(\sigma)\right\vert})

of nn given by the lengths of its permutation cycles (this Prop.).

Given any such partition (l 1l 2l k)(l_1 \geq l_2 \geq \cdots \geq l_k), we consider the following symmetric polynomial in nn variables:

(7)p (l 1l 2l k)(x 1,x 2,,x n)(x 1 l 1++x n l 1)(x 1 l 2++x n l 2)(x 1 l k++x n l k). p_{ (l_1 \geq l_2 \geq \cdots \geq l_k) } (x_1, x_2, \cdots, x_n) \;\;\;\coloneqq\;\;\; \big( x_1^{l_1} + \cdots + x_n^{l_1} \big) \cdot \big( x_1^{l_2} + \cdots + x_n^{l_2} \big) \cdots \big( x_1^{l_k} + \cdots + x_n^{l_k} \big) \,.


(Frobenius formula)

For nn \in \mathbb{N} and λ\lambda a partition of nn, the Schur polynomial s λs_\lambda is equal to the sum over permutations σ\sigma of the character values χ (λ)\chi^{(\lambda)} (5) at σ\sigma times the symmetric polynomial (7) which is indexed by the cycle lengths (6) of σ\sigma:

s λ=1n!σSym(n)χ (λ)(σ)p CycLengths(σ). s_\lambda \;=\; \frac{1}{n!} \underset {\sigma \in Sym(n)} {\sum} \chi^{(\lambda)}(\sigma) \cdot p_{CycLengths(\sigma)} \,.

(Sagan 01, Thm. 4.6.4, review in Sagan Enc., Thm. 3)

Schur-Weyl duality

The Schur polynomials are precisely the irreducible characters of finite dimensional polynomial representations of GL(n).

Also, the character χ λ\chi_{\lambda} of V λV_\lambda, the irreducible representation of S kS_k attached to λ\lambda (for kk the size |λ||\lambda| of the partition) maps to the Schur polynomial under the character map chch from virtual characters to symmetric polynomials.

This correspondence between linear representations of the symmetric groups and the general linear groups is called Schur-Weyl duality.

Generalizations via Schur functors

Schur functors may be viewed as a categorification of Schur functions. In fact, the Schur functors make sense in more general symmetric monoidal categories than VectorSpaces. It is a theorem in the case of vector space that the trace of

a Schur functor S λ(V)S λ(g)S λ(V)\mathbf{S}_\lambda(V)\stackrel{\mathbf{S}_\lambda(g)}\to \mathbf{S}_\lambda(V) on an endomorphism gGL(V)g\in GL(V) is the Schur function of the eigenvalues of gg. Considering the trace of a Schur functor makes sense in a general situation allowing for Schur functors and for the trace (rigid monoidal category); of course choosing appropriate variables to express the trace may depend on a context.



The concept first appears in work by Carl Jacobi on determinants.

It is named after:

  • Issai Schur, Über eine Klasse von Matrizen die sich einer gegeben Matrix zuordnen lassen, Inaugural-Dissertation, Berlin (1901) JMF 32.0165.04

See also

Textbook accounts:


See also:

  • Olivier Blondeau-Fournier, Pierre Mathieu, Schur superpolynomials: combinatorial definition and Pieri rule, arxiv/1408.2807

  • Miles Jones, Luc Lapointe, Pieri rules for Schur functions in superspace, arxiv/1608.08577

Last revised on May 23, 2021 at 17:46:28. See the history of this page for a list of all contributions to it.