# Specht module

A Specht module is a construction in the representation theory of the symmetric group \( S_n \): for each partition \( \lambda \) of \( n \), the Specht module \( S^{\lambda} \) is a module for \( S_n \) built from Young tableaux of shape \( \lambda \), and over a field of characteristic zero these modules give exactly the irreducible representations of \( S_n \). They were introduced in 1935 by [Wilhelm Specht](https://www.edgepedia.org/wilhelm-specht).<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( S^{\lambda} = \operatorname{span}\{ e_t : t \text{ a } \lambda\text{-tableau} \} \), a submodule of the permutation module \( M^{\lambda} \)<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> |
| Origin | Introduced in 1935 by W. Specht<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> |
| Characteristic 0 | When \( k \) contains \( \mathbf{Q} \), each \( S^{\lambda} \) is simple, and the \( S^{\lambda} \) for all partitions \( \lambda \) of \( n \) form a full set of simple \( kS_n \)-modules<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> |
| Dimension | Equal to the number of standard \( \lambda \)-tableaux, computable by the hook length formula<sup>[2](https://link.springer.com/article/10.1007/s00454-020-00208-w)</sup><sup> • </sup><sup>[3](https://www.math.columbia.edu/~calebji/teaching/schur-weyl.pdf)</sup> |
| Characteristic p | Specht modules are no longer always simple; for p-regular \( \lambda \), \( S^{\lambda} \) has a unique simple quotient \( D^{\lambda} \), and these \( D^{\lambda} \) form a full set of simple \( kS_n \)-modules<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> |
| Sign twist | \( S^{\lambda} \otimes \operatorname{sgn} \cong S^{\lambda'} \), where \( \lambda' \) is the conjugate partition<sup>[4](https://link.springer.com/article/10.1007/s00209-024-03646-x)</sup> |
| Basis | A characteristic-free basis (the Specht basis) is indexed by the standard \( \lambda \)-tableaux<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> |

## Construction from tableaux

A \( \lambda \)-tableau is a filling of the [Young diagram](https://www.edgepedia.org/young-diagram) of shape \( \lambda \) with the numbers \( 1, \dots, n \). A tabloid is an equivalence class of tableaux in which the order of entries within each row is ignored; the tabloids span a permutation module \( M^{\lambda} \) on which \( S_n \) acts by permuting entries.<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup><sup> • </sup><sup>[3](https://www.math.columbia.edu/~calebji/teaching/schur-weyl.pdf)</sup>

**Polytabloids.** For a tableau \( t \), the column stabilizer \( C_t \) is the subgroup of \( S_n \) that permutes entries only within each column of \( t \). The polytabloid attached to \( t \) is the signed sum over this subgroup,

\[ e_t = \sum_{\pi \in C_t} \operatorname{sgn}(\pi) \, \{\pi t\}, \]

where \( \{\pi t\} \) is the tabloid of \( \pi t \) and \( \operatorname{sgn}(\pi) \) is the sign of the permutation.<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup><sup> • </sup><sup>[2](https://link.springer.com/article/10.1007/s00454-020-00208-w)</sup> The construction antisymmetrizes the tabloid in its columns: row permutations fix \( e_t \), while a column permutation \( c \) acts by \( c \cdot e_t = \operatorname{sgn}(c) e_t \).<sup>[5](https://math.uchicago.edu/~may/REU2025/REUPapers/Gaek.pdf)</sup>

The Specht module of shape \( \lambda \) is the span of all polytabloids \( e_t \) inside \( M^{\lambda} \). It is stable under \( S_n \), since \( \sigma e_t = e_{\sigma t} \) for every permutation \( \sigma \).<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> A small example shows the mechanism: for \( \lambda = (2,1) \) and the tableau with 1, 2 in the first row and 3 below, the column stabilizer is \( \{e, (13)\} \), so the polytabloid is the difference of two tabloids; the resulting module \( S^{(2,1)} \) is the standard two-dimensional irreducible representation of \( S_3 \).<sup>[6](https://www.symmetricfunctions.com/specht-modules.htm)</sup>

## Irreducibility in characteristic zero

The importance of Specht modules comes from a classification theorem. When the field \( k \) contains \( \mathbf{Q} \), each \( S^{\lambda} \) is a simple \( kS_n \)-module, and as \( \lambda \) ranges over the partitions of \( n \), the modules \( S^{\lambda} \) form a full set of simple \( kS_n \)-modules, one for each partition.<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> Over \( \mathbb{C} \) in particular, the Specht modules give a complete irredundant list of irreducible \( \mathbb{C}S_n \)-modules.<sup>[4](https://link.springer.com/article/10.1007/s00209-024-03646-x)</sup>

The proof of simplicity in characteristic zero uses the bilinear form induced on \( S^{\lambda} \) by the standard scalar product on the permutation module: in characteristic 0 one has \( S^{\lambda} \cap (S^{\lambda})^{\perp} = 0 \), which forces the module to be simple.<sup>[7](https://web.math.princeton.edu/~charchan/ModularRepresentationsSymmetricGroupSeminar)</sup>

**Dimension.** A \( \lambda \)-tableau is called standard if its entries increase along each row and down each column. The polytabloids \( e_t \) for standard \( t \) form a basis of \( S^{\lambda} \), so the dimension of \( S^{\lambda} \) equals the number of standard \( \lambda \)-tableaux.<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup><sup> • </sup><sup>[2](https://link.springer.com/article/10.1007/s00454-020-00208-w)</sup> This count is given explicitly by the [hook length formula](https://www.edgepedia.org/hook-length-formula),

\[ \dim S^{\lambda} = \prod_{(i,j) \in \lambda} \frac{n!}{h_{ij}}, \]

where \( h_{ij} \) is the hook length of the cell in row \( i \) and column \( j \) of the Young diagram.<sup>[3](https://www.math.columbia.edu/~calebji/teaching/schur-weyl.pdf)</sup>

## Modular behavior

When the field has positive characteristic \( p \), the Specht modules are no longer always simple, but they still organize the representation theory. A partition is p-regular when no part is repeated \( p \) or more times. For each p-regular \( \lambda \), the Specht module \( S^{\lambda} \) has a unique simple quotient \( D^{\lambda} \), and the modules \( D^{\lambda} \) for p-regular \( \lambda \) form a full set of simple \( kS_n \)-modules.<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup><sup> • </sup><sup>[8](https://www.math.rwth-aachen.de/~Frank.Luebeck/data/SpechtGram/index.html?LANG=en)</sup> Determining the dimensions of these modules \( D^{\lambda} \) was described as a major open problem in the 1999 Encyclopedia of Mathematics article.<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup>

The natural bilinear form on \( S^{\lambda} \) carries modular information. Its [Gram matrix](https://www.edgepedia.org/gram-matrix) has elementary divisors, and the dimension of \( D^{\lambda} \) equals the number of elementary divisors not divisible by \( p \), that is, the rank of the Gram matrix reduced modulo \( p \). The divisors divisible by powers of \( p \) define the Jantzen filtration \( S = S(0) \ge S(1) \ge \cdots \) of the Specht module, and a theorem of Jantzen and Schaper gives a combinatorial formula for the determinant of the Gram matrix.<sup>[8](https://www.math.rwth-aachen.de/~Frank.Luebeck/data/SpechtGram/index.html?LANG=en)</sup> The Schaper number \( \nu_p(\lambda) \), the highest power of \( p \) dividing all inner products \( \langle x, y \rangle \) of polytabloids, satisfies \( \nu_p(\lambda) \ge 1 \) exactly when \( \lambda \) is p-singular, and knowledge of Schaper numbers yields upper bounds on decomposition numbers \( [S^{\lambda} : D^{\mu}] \).<sup>[9](https://academic.oup.com/qjmath/article/73/1/127/6249566)</sup>

## Branching and related structures

**Branching rule.** In characteristic 0, restricting \( S^{\lambda} \) to the subgroup \( S_{n-1} \) decomposes as a direct sum \( \bigoplus_{\mu} S^{\mu} \), where \( \mu \) runs over the partitions of \( n-1 \) obtained from \( \lambda \) by removing a single node from its Young diagram. In characteristic \( p \), Peel showed in 1971 that the restriction has a filtration whose successive quotients are the \( S^{\mu} \) for the same set of \( \mu \), although the detailed structure of the restriction is harder to determine.<sup>[10](https://archive.maths.nuim.ie/staff/jmurray/Preprints/branch.pdf)</sup>

**Sign twist and alternating groups.** Tensoring \( S^{\lambda} \) with the sign representation gives a module isomorphic to the dual \( (S^{\lambda'})^{*} \), where \( \lambda' \) is the conjugate partition, and over a field of characteristic zero this dual is unnecessary, so \( S^{\lambda} \otimes \operatorname{sgn} \cong S^{\lambda'} \) holds there.<sup>[12](https://www.cambridge.org/core/journals/bulletin-of-the-london-mathematical-society/article/the-irreducible-specht-modules-in-characteristic-2/98DBC6CACBE8165B06FD45761F0CC533)</sup> This relation underlies the description of the irreducible representations of the alternating group \( \mathfrak{A}_n \): when \( \lambda \neq \lambda' \), the restriction of \( S^{\lambda} \) to \( \mathfrak{A}_n \) remains irreducible, while for self-conjugate \( \lambda \) it splits into a pair of irreducible \( \mathfrak{A}_n \)-modules.<sup>[4](https://link.springer.com/article/10.1007/s00209-024-03646-x)</sup>

**Specht polynomials.** A classical construction realizes \( S^{\lambda} \) inside a polynomial ring. To a standard [Young tableau](https://www.edgepedia.org/young-tableau) \( T_{\lambda} \) one associates the monomial \( X^{t_{\lambda}} = \prod_{i=1}^{n} X_i^{m(i)-1} \), where \( m(i) \) is the row of \( T_{\lambda} \) containing \( i \), and the Specht polynomial \( sp_{T_{\lambda}} \) is the product over columns of Vandermonde determinants \( \prod_{j<l} (X_{\mathcal{C}_i(j)} - X_{\mathcal{C}_i(l)}) \), equivalently a signed sum over the column stabilizer. The Specht polynomials for the standard tableaux of shape \( \lambda \) span an \( S_n \)-submodule of \( \mathbb{R}[X] \) isomorphic to \( S^{\lambda} \).<sup>[2](https://link.springer.com/article/10.1007/s00454-020-00208-w)</sup> This polynomial model also unifies with the modular theory: symmetrized Specht polynomials give a practical way of evaluating the modular irreducible representations corresponding to p-regular partitions, an approach implemented in the computer algebra system SYMMETRICA.<sup>[11](https://www.sciencedirect.com/science/article/pii/S0195669803001951)</sup>

**Schur–Weyl duality.** The representation theory of \( S_n \) is tied to that of the general linear groups. Under this correspondence, Specht modules play for \( S_n \) the role that Weyl modules play for \( \operatorname{GL}_n \).<sup>[1](https://encyclopediaofmath.org/wiki/Specht_module)</sup> Concretely, Schur–Weyl duality decomposes the tensor power \( V^{\otimes k} \) of the standard representation of \( \operatorname{GL}(n, \mathbb{C}) \) as \( \bigoplus_{|\lambda|=k} V_{\lambda} \otimes S^{\lambda} \), pairing each irreducible of \( \operatorname{GL}(n, \mathbb{C}) \) with a Specht module of the same partition label.<sup>[3](https://www.math.columbia.edu/~calebji/teaching/schur-weyl.pdf)</sup>

## References

1. [Specht module, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Specht_module)
2. [Symmetric Non-Negative Forms and Sums of Squares, Discrete & Computational Geometry (Springer)](https://link.springer.com/article/10.1007/s00454-020-00208-w)
3. [Caleb Ji, Schur–Weyl Duality (lecture notes)](https://www.math.columbia.edu/~calebji/teaching/schur-weyl.pdf)
4. [Representations of symmetric and alternating groups and their double covers that remain irreducible modulo every prime, Mathematische Zeitschrift (Springer)](https://link.springer.com/article/10.1007/s00209-024-03646-x)
5. [Construction of Irreducible Representations of the Symmetric Group, REU paper, University of Chicago](https://math.uchicago.edu/~may/REU2025/REUPapers/Gaek.pdf)
6. [Specht modules, SymCat](https://www.symmetricfunctions.com/specht-modules.htm)
7. [Modular Representations of Symmetric Groups, seminar notes, Princeton University](https://web.math.princeton.edu/~charchan/ModularRepresentationsSymmetricGroupSeminar)
8. [Frank Lübeck, Elementary divisors of Gram matrices for Specht modules of symmetric groups and Jantzen filtrations](https://www.math.rwth-aachen.de/~Frank.Luebeck/data/SpechtGram/index.html?LANG=en)
9. [On the Schaper Numbers of Partitions, The Quarterly Journal of Mathematics (Oxford Academic)](https://academic.oup.com/qjmath/article/73/1/127/6249566)
10. [Harald Ellers and John Murray, Branching Rules for Specht Modules](https://archive.maths.nuim.ie/staff/jmurray/Preprints/branch.pdf)
11. [Modular irreducible representations of the symmetric group as linear codes, European Journal of Combinatorics (ScienceDirect)](https://www.sciencedirect.com/science/article/pii/S0195669803001951)
12. [THE IRREDUCIBLE SPECHT MODULES IN CHARACTERISTIC 2, Bulletin of the London Mathematical Society, Cambridge Core](https://www.cambridge.org/core/journals/bulletin-of-the-london-mathematical-society/article/the-irreducible-specht-modules-in-characteristic-2/98DBC6CACBE8165B06FD45761F0CC533)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Computational and symbolic algebra › Algebraic combinatorics and graph theory › Young tableaux and representations of the symmetric group*

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