nLab Schur function

Idea

The Schur polynomials $s_\lambda$, are polynomials in $n$ variables indexed by partitions $\lambda = \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$ of $n$, which constitute a linear basis of the ring of symmetric polynomials in $n$ variables

Definition

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

$$a_\alpha \;\coloneqq\; det\big(x_i^{\alpha_j}\big).$$

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

$$s_{\lambda}(x_1,x_2,\dots,x_n) = a_{\lambda+\delta}/a_{\delta}.$$

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 $s_{\lambda}$ for each partition, depending on countably many variables $x_1, x_2,\dots,x_n,x_{n+1}, \dots$.

Properties

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:

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

Given a semistandard Young tableau $T$, we write $X^T$ for the monomial which contains one factor of the variable $x_k$ for each occurrence of $k$ in the Young tableau:

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

We write

(2)$$ssYT_\lambda \;\supset\; ssYY_\lambda(N)$$

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

Proposition

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

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

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

This means that the Schur polynomial $s_\lambda$ in $N$ variables is the sum over semistandard Young tableaux (2) of shape $\lambda$ and with labels bounded as $T_{i,j} \leq N$:

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

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

(4)$$s_\lambda \big( x_1\!=\!1, \cdots, x_N \!=\! 1 \big) \;\;=\;\; \underset{ {T \in ssYT_\lambda(N)}, }{\sum} 1 \;\;=\;\; \left\vert ssYT_\lambda(N) \right\vert \,.$$

An immediate consequence is:

Proposition

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

In terms of $Sym(n)$-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 $n \in \mathbb{N}$ is a weakly decreasing sequence

$$\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)$$

of natural numbers whose sum equals $n$:

$$\lambda_1 + \lambda_2 + \cdots + \lambda_k \;=\; n \,.$$

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

(5)$$\chi^{(\lambda)}(-) \;\coloneqq\; Tr_{Specht^{(\lambda)}}(-)$$

for the irreducible character corresponding to this Specht module.

We write

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

for the function that sends a permutation $\sigma$ of $n$ elements to the partition

(6)$$CycLengths(\sigma) \;=\; (l_1 \geq l_2 \geq \cdots \geq l_{\left\vert Cycles(\sigma)\right\vert})$$

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

Given any such partition $(l_1 \geq l_2 \geq \cdots \geq l_k)$, we consider the following symmetric polynomial in $n$ variables:

(7)$$p_{ (l_1 \geq l_2 \geq \cdots \geq l_k) } (x_1, x_2, \cdots, x_n) \;\;\;\coloneqq\;\;\; \big( x_1^{l_1} + \cdots + x_n^{l_1} \big) \cdot \big( x_1^{l_2} + \cdots + x_n^{l_2} \big) \cdots \big( x_1^{l_k} + \cdots + x_n^{l_k} \big) \,.$$

Proposition

(Frobenius formula)

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

$$s_\lambda \;=\; \frac{1}{n!} \underset {\sigma \in Sym(n)} {\sum} \chi^{(\lambda)}(\sigma) \cdot p_{CycLengths(\sigma)} \,.$$

(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 $\chi_{\lambda}$ of $V_\lambda$, the irreducible representation of $S_k$ attached to $\lambda$ (for $k$ the size $|\lambda|$ of the partition) maps to the Schur polynomial under the character map $ch$ 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 $\mathbf{S}_\lambda(V)\stackrel{\mathbf{S}_\lambda(g)}\to \mathbf{S}_\lambda(V)$ on an endomorphism $g\in GL(V)$ is the Schur function of the eigenvalues of $g$. 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.

Related concepts

• Schur positivity

• representation theory of the symmetric group

Generalizations:

• Jack polynomial

• Macdonald polynomial

• noncommutative Schur function

• quasisymmetric Schur function

References

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

• Richard Stanley, Section II of: Theory and application of plane partitions 1, Studies in Applied Math. 50 2 (1971), 167-188 (pdf, StanleyPlanePartitions1.pdf)

Textbook accounts:

• Ian G. Macdonald, Section I.3 of: Symmetric functions and Hall polynomials, Oxford Math. Monographs, 2nd enlarged ed. 1995 (ISBN:9780198739128)

• Richard Stanley, Sections 7.10, 7.15 in: Enumerative combinatorics 2, Cambridge University Press (1999, 2010) (doi:10.1017/CBO9780511609589, webpage)

• Bruce Sagan, Section 4.4 of: The symmetric group, Springer 2001 (doi:10.1007/978-1-4757-6804-6, pdf)

• Stuart Martin, Schur algebras and representation theory, Cambridge Univ. Press 1994

Survey:

• Bruce E. Sagan, Schur functions, in: Michiel Hazewinkel (ed.), Encyclopaedia of Mathematics, Springer 2002 (pdf, pdf)

• Wikipedia, Schur polynomial

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