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The 3-cycle filter for ν=(3,2) on S_8

Example

For $ν = (3, 2) ⊢ 5$ and $r = 3$ the filter The $r$ -cycle virtual character filter gives the virtual character of $S_{8}$ $Ψ_{(3, 2)} = χ^{(6, 2)} - χ^{(4, 4)} - χ^{(3, 2, 2, 1)} + χ^{(3, 2, 1, 1, 1)},$ because the four partitions $λ ⊢ 8$ with $λ / (3, 2)$ a removable $3$ -border strip are $(6, 2)$ (height $0$ ), $(4, 4)$ (height $1$ ), $(3, 2, 2, 1)$ (height $1$ ) and $(3, 2, 1, 1, 1)$ (height $2$ ), with signs $+, -, -, +$ . It vanishes on every class of $S_{8}$ whose cycle type has no part $3$ ; the computations below verify this on the classes without $3$ -cycles that are displayed in the statement: the identity class, where $Ψ_{(3, 2)}$ has degree $20 - 14 - 70 + 64 = 0$ , the class of $8$ -cycles, where all four characters vanish because none of the four shapes is a hook (The value of an irreducible character on a full cycle is supported on hooks), and the class $(4, 4)$ , where the rule gives $χ^{(6, 2)} = 0$ , $χ^{(4, 4)} = 2$ , $χ^{(3, 2, 2, 1)} = - 2$ , $χ^{(3, 2, 1, 1, 1)} = 0$ , so that $Ψ_{(3, 2)} = 0 - 2 - (- 2) + 0 = 0$ .

Facts & Assumptions

Given: The partitions $(3, 2) ⊢ 5$ and $(6, 2), (4, 4), (3, 2, 2, 1), (3, 2, 1, 1, 1) ⊢ 8$ , and for every partition of an integer the character $χ^{λ}$ of the complex Specht module $S^{λ}$ (Column antisymmetrizers, polytabloids, and Specht modules).

[F1]

Filter proposition: $Ψ_{(3, 2)} : = \sum_{λ / (3, 2)} (- 1)^{ht (λ / (3, 2))} χ^{λ} \in R (S_{8})$ is the outer product $χ^{(3, 2)} \circ ψ_{3}$ with $ψ_{3} = χ^{(3)} - χ^{(2, 1)} + χ^{(1, 1, 1)}$ , and it vanishes on every class of $S_{8}$ whose cycle type has no part $3$ ; $ψ_{3}$ vanishes off the class of $3$ -cycles, so $ψ_{3} (1) = 0$ (The $r$ -cycle virtual character filter).

[F2]

Removable border strip: a removable $3$ -strip of $λ$ is a connected, $2 \times 2$ -free skew diagram $λ / ν$ with $3$ nodes and $ν = (3, 2)$ ; its height is its number of rows minus one, equivalently $3 - c$ where $c$ is its number of columns (Removable border strips and their height, Hook, arm, leg, and hook length of a box).

[F3]

Murnaghan–Nakayama rule: $χ^{λ} (σ c) = \sum_{λ / μ} (- 1)^{ht (λ / μ)} χ^{μ} (σ)$ for a cycle $c$ of length $r$ on a specified set of points fixed by $σ$ (a specified fixed point for length $1$ ), summed over the removable $r$ -strips of $λ$ (The Murnaghan–Nakayama character rule).

[F4]

Full-cycle values: for an $8$ -cycle $ρ \in S_{8}$ , $χ^{λ} (ρ) = (- 1)^{8 - x}$ if $λ = (x, 1^{8 - x})$ , and $0$ otherwise; in particular $χ^{λ} (ρ) = 0$ whenever $[λ]$ contains a $2 \times 2$ block (The value of an irreducible character on a full cycle is supported on hooks).

[F5]

Dimensions by the hook length formula: $f^{(6, 2)} = 8! / (7 \cdot 6 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot 2 \cdot 1) = 20$ , $f^{(4, 4)} = 14$ , $f^{(3, 2, 2, 1)} = 70$ , $f^{(3, 2, 1, 1, 1)} = 64$ (The hook length formula).

[F6]

Class language: a class contains a $3$ -cycle exactly when its cycle type has a part $3$ , and the identity class is the class of cycle type $(1^{8})$ (Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type).

Verification

technique · direct

1.1F1F2givenalgebra

A partition $λ ⊢ 8$ contains $(3, 2)$ exactly when $λ_{1} \geq 3$ and $λ_{2} \geq 2$ , leaving the eleven candidates $(6, 2), (5, 2, 1), (4, 2, 2), (4, 2, 1, 1), (3, 2, 2, 1), (3, 2, 1, 1, 1), (5, 3), (4, 3, 1), (3, 3, 2), (3, 3, 1, 1), (4, 4)$ ; among their skew differences $λ / (3, 2)$ , which all have $3$ nodes, exactly four are edge-connected and contain no $2 \times 2$ block, namely the horizontal strip ${(1, 4), (1, 5), (1, 6)}$ with $c = 3$ columns in $λ = (6, 2)$ , the corners ${(1, 4), (2, 3), (2, 4)}$ and ${(3, 1), (3, 2), (4, 1)}$ with $c = 2$ in $λ = (4, 4)$ and $λ = (3, 2, 2, 1)$ , and the vertical strip ${(3, 1), (4, 1), (5, 1)}$ with $c = 1$ in $λ = (3, 2, 1, 1, 1)$ ; the other seven differences are disconnected, for instance $λ = (5, 3)$ , where the node $(2, 3)$ is not adjacent to $(1, 4)$ or $(1, 5)$ . The heights are $3 - c$ , namely $0, 1, 1, 2$ , so the four signs $+, -, -, +$ and the displayed expression for $Ψ_{(3, 2)}$ follow.

2.1step 1.1F1F5algebra

The four dimensions of [F5] give $Ψ_{(3, 2)} (1) = 20 - 14 - 70 + 64 = 0$ ; equivalently $Ψ_{(3, 2)} = χ^{(3, 2)} \circ ψ_{3}$ by [F1] and $ψ_{3} (1) = 0$ because the identity of $S_{3}$ is not a $3$ -cycle, so the degree vanishes.

2.2F4step 1.1algebra

On the class of $8$ -cycles each of the four shapes contains a $2 \times 2$ block (rows $1$ and $2$ , columns $1$ and $2$ ), so none is a hook and $χ^{λ} = 0$ on that class for all four $λ$ by the full-cycle lemma [F4]; hence $Ψ_{(3, 2)}$ vanishes there.

2.3step 1.1F3algebra

On the class $(4, 4)$ the rule [F3] gives: $χ^{(6, 2)} = 0$ , since the unique $4$ -strip takes $(6, 2)$ to $(2, 2)$ with height $0$ and $(2, 2)$ has no $4$ -strip; $χ^{(4, 4)} = + 1 - (- 1) = 2$ from the strips to $(4)$ (height $0$ ) and to $(3, 1)$ (height $1$ ), with $χ^{(4)} (4) = 1$ and $χ^{(3, 1)} (4) = - 1$ ; $χ^{(3, 2, 2, 1)} = (- 1) + (- 1) = - 2$ from the strips to $(3, 1)$ and to $(1^{4})$ , both of height $2$ , with $χ^{(3, 1)} (4) = χ^{(1^{4})} (4) = - 1$ ; and $χ^{(3, 2, 1, 1, 1)} = 0$ because $(3, 2, 1, 1, 1)$ has no $4$ -strip. Hence $Ψ_{(3, 2)} = + 0 - 2 - (- 2) + 0 = 0$ on the class $(4, 4)$ , whose cycle type has no part $3$ .

3.1step 2.1step 2.2step 2.3F1F6algebra∎

Steps 2.1, 2.2 and 2.3 verify the vanishing predicted by [F1] on the identity class, the class of $8$ -cycles and the class $(4, 4)$ , all of which are classes of $S_{8}$ without a $3$ -cycle, confirming the displayed value $Ψ_{(3, 2)} = 0$ there.

Remarks

Consistency of the four shapes with the hook-length list. The removable $3$ -strips found in step 1.1 correspond to the hook lengths of the four shapes: for instance the strip to $(6, 2)$ is the hook of the node $(1, 4)$ , of length $3$ ; the full hook-length multiset of $(6, 2)$ is $7, 6, 4, 3, 2, 1, 2, 1$ , and the other strips arise from further hooks of length $3$ in the respective shapes.

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Sources

G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer 1978 (complete author-hosted scan, 161 pp.) (standard reference, not scraped)
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Mathematical Monographs, 1995 (complete author-hosted scan, 486 pp.) (standard reference, not scraped)