Math 202B: Lecture 25

Let n \in \mathbb{N} be arbitrary but fixed, and let \Lambda_n denote the set of integer partitions of n, or equivalently the set of Young diagrams with exactly n cells. For each \lambda \in \Lambda_n, let (V^\lambda,\varphi^\lambda) be a representative of the corresponding isomorphism class of irreducible unitary representations of the symmetric group S_n=\mathrm{Aut}X, where X=\{1,\dots,n\} is a set of n-points. Equivalently, (V^\lambda,\Phi^\lambda) is a representative of the equivalence class of irreducible linear representations of the convolution algebra \mathcal{C}(S_n). We will write partitions \lambda \in \Lambda_n as vectors \lambda=(\lambda_1,\dots,\lambda_r) with 1 \leq r \leq n entries, which are weakly decreasing positive integers whose sum is n.

In Lectures 23 and 24 we constructed three irreducible unitary representations of S_n by hand: the trivial and alternating representations, which are one-dimensional, and the standard representation, which is (n-1)-dimensional. An important point is that there is no canonical way to say which of the abstract irreducible representations (V^\lambda,\varphi^\lambda) we should declare to be the standard (or alternating, or trivial) representation. More generally, if we somehow construct another concrete irreducible representation of S_n, then there is no obvious way to say which of the abstractly defined irreducible representations (V^\lambda,\varphi^\lambda). The main result in the representation theory of the symmetric groups is that it is possible to give a concrete construction of each (V^\lambda,\varphi^\lambda) such that the following holds.

Theorem 25.1. (Branching rule) For any \lambda \in \Lambda_n, when we view the irreducible representation (V^\lambda,\varphi^\lambda) of S_n as a representation of S_{n-1}, it becomes a reducible representation whose isotypic decomposition is

V^\lambda = \bigoplus\limits_{\mu \nearrow \lambda} V^\mu,

where the sum is over all \mu \in \Lambda_{n-1} such that \lambda can be obtained from \mu by adding a single cell.

For example, when the irreducible representation V^{(2,2)} of S_4 is viewed as a representation of S_3, where we identify S_3 with the subgroup of S_4 consisting of permutations which have 4 as a fixed point, its decomposition into irreducible representations of S_3 is

V^{(2,2)} \simeq V^{(2,1)},

i.e. it remains irreducible and is isomorphic to the standard representation of S_3. If we instead start with V^{(2,1,1)}, then as a representation of S_3 we have

V^{(2,1,1)} \simeq V^{(2,1)} \oplus V^{(1,1,1)},

the direct sum of the standard and alternating representations.

The main consequence of the branching rule is that the representation theory of the symmetric groups is controlled by an infinite partially ordered set \Lambda called Young’s lattice, which as a set is the disjoint union,

\Lambda= \bigsqcup\limits_{n=1}^\infty \Lambda_n,

i.e. the set of all Young diagrams, with the partial order defined by

\mu \leq \lambda \iff \lambda \text{ can be obtained from }\mu \text{ by adding cells}.

As a consequence, if (V^\lambda,\varphi^\lambda) is any irreducible representation of S_n, then by restricting (n-1) times we get a decomposition of V^\lambda as a direct sum of one-dimensional subspaces, and choosing a unit vector in each gives a basis

\{e_T \colon T \in \mathrm{SYT}(\lambda)\}

of V^\lambda indexed by the set of standard Young tableaux of shape \lambda. These combinatorial objects can either be viewed as increasing paths in the Hasse graph of Young’s lattice from the bottom unicellular diagram up to \lambda in which we add one cell at a time, or as ways of writing the numbers 1,\dots,n in the cells of \lambda such that the numbering increases along rows in columns, thereby encoding such a growth process. Thus as a consequence of the branching rule, we have the following.

Theorem 25.2. The dimension of V^\lambda equals the number of Standard Young Tableaux of shape \lambda.

There is in fact a combinatorial formula for the number of standard Young tableaux of a given shape: it is called the hook length formula, and at least for relatively small Young diagrams it is a useful way to count standard Young tableaux.

More ambitiously, instead of just knowing the dimension of V^\lambda, one would like to have an explicit form for the matrices of the operators \varphi^\lambda(g), g \in S_n, with respect to the branching basis \{e_T \colon T \in \mathrm{SYT}(\lambda)\}. A related question is to calculate the character of this representation, i.e. the function on S_n given by

\chi^\lambda(g) = \mathrm{Tr}\, \rho^\lambda(g), \quad g \in S_n.

Equivalent to this is the problem of computing the central characters \omega^\lambda_\alpha of the class algebra \mathcal{Z}(S_n), where we recall that \omega^\lambda_\alpha is the eigenvalue of an element K_\alpha of the class basis \{K_\alpha \colon \alpha \in \Lambda_n\}. This is an equivalent problem because

\omega_\alpha^\lambda = |C_\alpha|\frac{\chi^\lambda_\alpha}{\dim V^\lambda},

where \chi^\lambda_\alpha is the value of \chi^\lambda(g) on any permutation g \in S_n belonging to the conjugacy class C_\alpha \subseteq S_n. For any Young diagram \lambda and any cell \Box \in \Lambda, define the content c(\Box) of this cell to be its column index minus its row index.

Theorem 25.3. The central character of the conjugacy class of transpositions acting in the irreducible representation V^\lambda is the sum of the contents of the Young diagram \lambda,

\omega^{\lambda}_{(2,1,\dots,1)} = \sum\limits_{\Box \in \lambda} c(\Box).

Published by Jonathan Novak

Mathematician at UC San Diego.

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