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Polytabloids of shape $(2, 1)$

Statement

In $M^{(2, 1)}$ , let $v_{i}$ be the tabloid whose second row is $i$ , for $i = 1, 2, 3$ . For $t = \begin{matrix} 1 & 2 \\ 3 \end{matrix}$ and $u = \begin{matrix} 1 & 3 \\ 2 \end{matrix}$ , the standard polytabloids are $e_{t} = v_{3} - v_{1}$ and $e_{u} = v_{2} - v_{1}$ . Every $(2, 1)$ -polytabloid is one of $\pm (v_{3} - v_{1})$ , $\pm (v_{2} - v_{1})$ , and $\pm (v_{3} - v_{2})$ , and $e_{t}, e_{u}$ form a basis of $S^{(2, 1)}$ .

Facts & Assumptions

Given: Work over $C$ with the shape $λ = (2, 1)$ and entries ${1, 2, 3}$ .

[F1]

The tabloids form a basis of $M^{λ}$ (Young subgroups, tabloids, and permutation modules).

[F2]

Two tableaux define the same tabloid exactly when their row sets agree (Young subgroups, tabloids, and permutation modules).

[F3]

A tableau is standard when entries strictly increase along rows and down columns (Tableaux and standard tableaux).

[F4]

The column stabilizer consists of the permutations preserving each column set (Row and column stabilizers).

[F5]

The column antisymmetrizer is the signed sum over the column stabilizer, $e_{t} = κ_{t} \cdot {t}$ , and $S^{λ}$ is the span of all $λ$ -polytabloids (Column antisymmetrizers, polytabloids, and Specht modules).

[F6]

Sign is $(- 1)$ raised to the inversion number, and the Specht definition uses this sign after the canonical relabelling $i \mapsto i - 1$ (Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations, Column antisymmetrizers, polytabloids, and Specht modules).

[F7]

For a partition of $n$ , the standard polytabloids form a basis of the complex Specht module (Standard polytabloids form a basis of a complex Specht module).

No form of the Axiom of Choice is used; the calculation explicitly lists a finite set of tableaux.

Proof

technique · direct

1.1givenF1F2

The three tabloids are $v_{1}, v_{2}, v_{3}$ : the second row is a singleton, and its label determines the first row as the complementary pair. They are distinct by [F2], so they are exactly the tabloid basis of [F1].

1.2givenF4F5F6

Write $[a b; c]$ for a tableau with first row $a, b$ and second row $c$ , where ${a, b, c} = {1, 2, 3}$ . Its columns are ${a, c}$ and ${b}$ , so [F4] gives $C_{s} = {1, (a c)}$ . The three transpositions, in one-line notation on the labels $1, 2, 3$ , are $213$ , $321$ , and $132$ , with respectively $1$ , $3$ , and $1$ inversions, and the order-preserving relabelling ${1, 2, 3} \to {0, 1, 2}$ preserves these counts. Thus [F6] gives $sgn (a c) = - 1$ , and [F5] yields $κ_{s} = 1 - (a c)$ . The second row of ${s}$ is ${c}$ , so applying $(a c)$ changes it to ${a}$ and $e_{s} = v_{c} - v_{a}$ .

2.1givenstep 1.2

Applying step 1.2 to all six tableaux gives $e_{[1 2; 3]} = v_{3} - v_{1}$ , $e_{[2 1; 3]} = v_{3} - v_{2}$ , $e_{[1 3; 2]} = v_{2} - v_{1}$ , $e_{[3 1; 2]} = v_{2} - v_{3}$ , $e_{[2 3; 1]} = v_{1} - v_{2}$ , and $e_{[3 2; 1]} = v_{1} - v_{3}$ . These are precisely the three listed differences and their negatives.

2.2givenF1F3step 1.1step 1.2algebra

The row and column inequalities in [F3] leave exactly $t = [1 2; 3]$ and $u = [1 3; 2]$ as standard tableaux. Their polytabloids $v_{3} - v_{1}$ and $v_{2} - v_{1}$ are linearly independent: in a relation $α (v_{3} - v_{1}) + β (v_{2} - v_{1}) = 0$ , the coefficients of the distinct basis vectors $v_{3}$ and $v_{2}$ force $α = β = 0$ .

3.1F7step 2.1step 2.2∎

By [F7], the standard polytabloids of shape $(2, 1)$ form a basis of $S^{(2, 1)}$ ; step 2.2 identifies that standard family as exactly $e_{t}, e_{u}$ . Together with the six explicit calculations in step 2.1, this proves the Statement.

Depends on

Used by

All three Specht modules of S₃ Example

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charlotte Chan, Representation Theory of Symmetric Groups, Remark 3.9 and its explicit shape-(2,1) polytabloid calculations, printed p. 12 (standard reference, not scraped)
Mark Wildon, Representation Theory of the Symmetric Group, Definition 2.4 and Example 2.6(B), printed pp. 5-6 (standard reference, not scraped)

Polytabloids of shape (2,1)(2,1)