Schur algebra ( $S (m, n) = End_{C S_{n}} ((C^{m})^{\otimes n})$ )

The Schur algebra is the commutant of place permutations on a tensor power. It is the linear span of the diagonal general linear group action, by Schur–Weyl duality. Modules over

S (m, n)

correspond to homogeneous degree-

n

polynomial representations. Over the complex numbers it is a semisimple algebra: decompose the tensor power as a module for the semisimple group algebra

C S_{n}

, and take its endomorphism algebra.

Table of contents
- Invertible tensor powers span the Schur algebra Schur algebra
- Schur–Weyl duality Schur algebra
  - Trace of a permuted tensor power Schur–Weyl duality
  - Young-symmetrizer tensor decomposition Schur–Weyl duality

Invertible tensor powers span the Schur algebra ( $S (m, n) = span {g^{\otimes n} : g \in GL (V)}$ )

The commutant of the permutation action on

V^{\otimes n}

identifies with the symmetric tensors in

End (V)^{\otimes n}

. Polarization spanning of symmetric tensors spans it by

a^{\otimes n}

. Interpolating

(a + t I)^{\otimes n}

at

n + 1

values where

a + t I

is invertible expresses each such power using invertible ones. Hence the Schur algebra is the linear span of the diagonal general linear action.

Schur–Weyl duality ( $V^{\otimes n} = ⨁_{λ ⊢ n} S^{λ} \otimes D_{λ} (V)$ )

The commuting actions of the symmetric group and the general linear group on a tensor power are mutual commutants. The displayed sum ranges over partitions with at most

dim V

rows. The Specht modules and Schur modules are the simple factors for the two actions. In particular a primitive Young symmetrizer selects one copy of the matching Schur module.

Trace of a permuted tensor power ( $tr (σ ξ^{\otimes n}) = \prod_{r} tr (ξ^{r})^{α_{r}}$ )

If

σ

has

α_{r}

cycles of length

r

, contraction of matrix entries around its cycles gives

tr (σ ξ^{\otimes n}) = \prod_{r} tr (ξ^{r})^{α_{r}}

. The formula holds for every endomorphism, not only diagonalizable ones. The Schur–Weyl duality decomposition equates it with a sum of products of Specht and Schur characters.

Young-symmetrizer tensor decomposition ( $V^{\otimes n} = ⨁_{λ, t standard} h_{t} V^{\otimes n}$ )

The standard-tableau right-ideal decomposition of

C S_{n}

, tensored over that algebra with

V^{\otimes n}

, gives this direct sum of general linear group modules. With

e_{t} = h_{t} / H_{λ}

, the map

e_{t} C S_{n} \otimes T \to e_{t} T

is an isomorphism. Individual summands generally need not be symmetric group submodules; instead each is a Schur module for the commuting action.

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