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The value of an irreducible character on a full cycle is supported on hooks

Statement

Let $n \geq 1$ and let $ρ \in S_{n}$ be a cycle of length $n$ (for $n = 1$ , the identity on the sole point) (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation). Then for every partition $λ ⊢ n$ , taking $λ_{2} = 0$ for a one-row partition, with $χ^{λ}$ the character of the complex Specht module $S^{λ}$ (Column antisymmetrizers, polytabloids, and Specht modules, Virtual characters and the character ring $R (G)$ of a finite group),

$χ^{λ} (ρ) = {\begin{cases} (- 1)^{n - x}, & λ = (x, 1^{n - x}) for some 1 \leq x \leq n, \\ 0, & otherwise, \end{cases}$

where $(x, 1^{n - x})$ denotes the hook partition of $n$ with first part $x$ . In particular $χ^{(n)} (ρ) = 1$ and $χ^{(1^{n})} (ρ) = (- 1)^{n - 1}$ , and $χ^{λ} (ρ) = 0$ whenever $[λ]$ contains a $2 \times 2$ block of nodes, equivalently whenever $λ_{2} \geq 2$ ; in the hook case $λ = (x, 1^{n - x})$ the numbers of rows and columns are $n - x + 1$ and $x$ . No choice principle is used.

Facts & Assumptions

Given: An integer $n \geq 1$ , a cycle $ρ \in S_{n}$ of length $n$ , and a partition $λ = (λ_{1} \geq \dots \geq λ_{k} > 0)$ of $n$ with $k$ parts.

[F1]

Determinantal form: $χ^{λ} = \det ([λ_{i} - i + j]) = \sum_{w} sgn (w) [λ_{1} - 1 + w (1)] \circ [λ_{2} - 2 + w (2)] \circ \dots \circ [λ_{k} - k + w (k)]$ in $R (S_{n})$ , the sum running over bijections $w$ of ${1, \dots, k}$ , where $[m] \in R (S_{m})$ is the trivial character for $m \geq 0$ and $[m] : = 0$ for $m < 0$ , and $\circ$ is the outer induction product (The determinantal form of a Specht character, The outer induction product of symmetric-group characters).

[F2]

For $m_{1}, \dots, m_{k} \geq 0$ the product $[m_{1}] \circ \dots \circ [m_{k}]$ is the character of the permutation module on tabloids of shape $μ$ , where $μ$ is the partition obtained by discarding the zero entries and reordering. Indeed the characteristic isomorphism sends this product to $\prod_{i} h_{m_{i}} = h_{μ}$ (step 1.1 of The determinantal form of a Specht character and The Frobenius characteristic is an isometric graded ring isomorphism); the tabloid character has the same characteristic by The characteristic of a Young permutation character is complete homogeneous, so injectivity identifies them. Negative entries give the zero character, and zero entries give unit factors. The value of a permutation character is the number of fixed points (The character of a permutation representation counts fixed points).

[F3]

The character $χ^{λ}$ , like every element of $R (S_{n})$ , is a complex-valued class function on $S_{n}$ , and evaluation at $ρ$ is linear on $R (S_{n})$ (Virtual characters and the character ring $R (G)$ of a finite group).

[F4]

A cycle of length $k$ has sign $(- 1)^{k - 1}$ (A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles).

Proof

technique · direct

1.1F2given

For integers $m_{1}, \dots, m_{k}$ with $\sum_{i} m_{i} = n$ , the value of $[m_{1}] \circ \dots \circ [m_{k}]$ at the $n$ -cycle $ρ$ is $1$ if exactly one $m_{i}$ equals $n$ and all other $m_{i}$ equal $0$ , and is $0$ otherwise. Indeed, if some $m_{i}$ is negative the product is $0$ ; otherwise it is the character of the permutation module on tabloids of shape $μ$ , the partition of the positive entries, and its value at $ρ$ counts the tabloids fixed by $ρ$ . A tabloid is an ordered tuple of nonempty blocks of sizes $μ_{1}, μ_{2}, \dots$ and it is fixed exactly when $ρ$ preserves each block as a set; a subset of ${1, \dots, n}$ is invariant under the $n$ -cycle $ρ$ only if it is empty or all of ${1, \dots, n}$ . So a fixed tabloid has a single nonempty block, i.e. $μ = (n)$ and the counted value is $1$ ; if $μ \neq (n)$ the count is $0$ . Since the positive entries of $(m_{1}, \dots, m_{k})$ form the partition in question, the value is $1$ precisely when the multiset ${m_{1}, \dots, m_{k}}$ is ${n, 0, \dots, 0}$ .

2.1step 1.1F1F3algebra

By [F1] and the linearity of evaluation [F3], $χ^{λ} (ρ) = \sum_{w} sgn (w) [λ_{1} - 1 + w (1)] \circ \dots \circ [λ_{k} - k + w (k)] (ρ)$ , and by step 1.1 the sum extends only over the bijections $w$ for which the multiset ${λ_{i} - i + w (i)}_{i = 1}^{k}$ equals ${n, 0, \dots, 0}$ . For such a $w$ , let $i_{0}$ be the unique index with $λ_{i_{0}} - i_{0} + w (i_{0}) = n$ ; then for every $j \neq i_{0}$ one has $λ_{j} - j + w (j) = 0$ , that is $w (j) = j - λ_{j}$ , and therefore $λ_{j} \leq j - 1$ for every $j \neq i_{0}$ .

3.1step 2.1F1F4algebra∎

We solve the condition of step 2.1. If $λ_{2} \geq 2$ , no bijection $w$ satisfies it: for $i_{0} = 1$ the index $j = 2$ would violate $λ_{j} \leq j - 1$ ; for $i_{0} = 2$ the index $j = 1$ would require $λ_{1} \leq 0$ ; and for $i_{0} \geq 3$ the index $j = 2$ would violate $λ_{2} \leq 1$ . Hence a surviving term exists only if $λ_{2} \leq 1$ , i.e. only for hooks $λ = (x, 1^{n - x})$ , where $k = n - x + 1$ . Conversely for a hook: for $j \geq 2$ one has $j - λ_{j} = j - 1 \in {1, \dots, k - 1}$ , these values are pairwise distinct, and for $j = 1$ the equation $m_{1} = n$ reads $w (1) = n - x + 1 = k$ ; so there is exactly one admissible $w$ , namely $w (1) = k$ and $w (j) = j - 1$ for $j \geq 2$ , a $k$ -cycle; it indeed satisfies $λ_{1} - 1 + w (1) = x - 1 + k = n$ and $λ_{j} - j + w (j) = 1 - j + j - 1 = 0$ for $j \geq 2$ . Its sign is $(- 1)^{k - 1} = (- 1)^{n - x}$ by [F4], so $χ^{λ} (ρ) = (- 1)^{n - x}$ for $λ = (x, 1^{n - x})$ , and $χ^{λ} (ρ) = 0$ whenever $λ_{2} \geq 2$ . The formula for $x = n$ gives $χ^{(n)} (ρ) = 1$ , that for $x = 1$ gives $χ^{(1^{n})} (ρ) = (- 1)^{n - 1}$ , and a hook $(x, 1^{n - x})$ has $n - x + 1$ rows and $x$ columns; a partition with $λ_{2} \geq 2$ is exactly one containing a $2 \times 2$ block, and it has at least two rows and its second row has at least two nodes.

Remarks

Consistency with the two extreme rows. For $x = n$ the hook is the single row $(n)$ and the value is $1$ , as it must be for the trivial character; for $x = 1$ it is the single column $(1^{n})$ and the value is $(- 1)^{n - 1}$ , the sign of an $n$ -cycle, as it must be for the sign character.
What fails for non-hooks. The value vanishes precisely when $λ$ contains a $2 \times 2$ block, equivalently when $λ_{2} \geq 2$ ; a hook with $x \geq 2$ and $n - x \geq 1$ such as $(2, 1)$ has several rows and columns and still has a nonzero value, so the vanishing criterion is the $2 \times 2$ block and not the plain row-and-column count. This is the sign fact behind the Murnaghan–Nakayama rule: the first hook removal in The Murnaghan–Nakayama character rule carries the sign $(- 1)^{ht}$ .

Depends on

Used by

Murnaghan–Nakayama values of the character of (5,4,4) Example
The 3-cycle filter for ν=(3,2) on S₈ Example
The S₄ character table by rim-hook recursion Example
The r-cycle virtual character filter Proposition
The Murnaghan–Nakayama character rule Theorem

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51 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

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)