What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?

Question

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.

The Weyl group $W(B_n)$ is the wreath product $C_2 \wr S_n = C_2^n \rtimes S_n$ of the cyclic group $C_2 = \{\pm 1\}$ and the symmetric group $S_n$. The group $W(B_n)$ is equipped with two "sign" homomorphisms:

• $sgn: W(B_n) \to \{\pm 1\}$ is trivial on $C_2^n$ and equal to the usual sign homomorphism on $S_n$.
• $\overline{sgn}: W(B_n) \to \{\pm 1\}$ is trivial on $S_n$ and acts on $C_2^n = \{\pm 1\}^n$ by multiplying the factors together.

The irreducible $W(B_n)$-modules are parameterized by bi-partitions $(\lambda,\mu)$ of $n$, i.e., ordered pairs of partitions $\lambda$ and $\mu$ such that $|\lambda|+|\mu| = n$. If $V^{(\lambda,\mu)}$ is the irreducible module labeled by the bi-partition $(\lambda,\mu)$, then one has

• $V^{(\lambda,\mu)} \otimes sgn \cong V^{(\lambda',\mu')}$, where $\lambda'$ and $\mu'$ are the partitions that are transpose to $\lambda$ and $\mu$, respectively, and
• $V^{(\lambda,\mu)} \otimes \overline{sgn} = V^{(\mu,\lambda)}$.

The Weyl group $W(D_n)$ is equal to the index-2 subgroup $\ker(\overline{sgn})$. This implies that:

1. If $\lambda \neq \mu$, then $V^{(\lambda,\mu)}$ and $V^{(\mu,\lambda)}$ are isomorphic and irreducible as $W(D_n)$-modules.
2. If $(\lambda,\lambda)$ is a bi-partition of $n$, then $V^{(\lambda,\lambda)}$ decomposes as a $W(D_n)$-module into a direct sum $V^{(\lambda,\lambda)^+} \oplus V^{(\lambda,\lambda)^-}$ of two non-isomorphic, conjugate, irreducible $W(D_n)$-modules.
3. Every irreducible $W(D_n)$-module is uniquely of one of the preceding forms, subject only to the $W(D_n)$-module isomorphism $V^{(\lambda,\mu)} \cong V^{(\mu,\lambda)}$ if $\lambda \neq \mu$.

The sign homomorphism $sgn$ restricts to a homomorphism on $W(D_n)$. What is the effect of tensoring with this sign representation on the irreducible $W(D_n)$-modules of the form $V^{(\lambda,\lambda)^+}$ and $V^{(\lambda,\lambda)^-}$?

If $\lambda \neq \lambda'$, then up to relabeling one could assume that $V^{(\lambda,\lambda)^+} \otimes sgn \cong V^{(\lambda',\lambda')^+}$ and $V^{(\lambda,\lambda)^-} \otimes sgn \cong V^{(\lambda',\lambda')^-}$, but what happens when $\lambda = \lambda'$? One must have either

• $V^{(\lambda,\lambda)^+} \otimes sgn \cong V^{(\lambda,\lambda)^+}$, or
• $V^{(\lambda,\lambda)^+} \otimes sgn \cong V^{(\lambda,\lambda)^-}$,

i.e., tensoring with $sgn$ either leaves the $W(D_n)$-constituents of $V^{(\lambda,\lambda)}$ fixed (up to isomorphism), or it swaps the constituents. Is there a known criterion, in terms of $\lambda$, for determining which situation is which?

Answer 1

Recall the construction of such representations. Let $H\subset S_{2n}$ be the subgroup fixing the partition (so the two parts can be swapped) $$\{1,\dots,n\}\sqcup\{1',\dots,n'\}.$$ It is isomorphic to the wreath product $S_n\wr S_2=S_n^2\rtimes S_2$, via the involution $\tau$ sending $i\mapsto i'$. In other words, elements of $H$ are of the form $(\sigma_1,\sigma_2)$ or $(\sigma_1,\sigma_2)\iota$ where $(\sigma_1,\sigma_2)$ denotes the permutation $$\begin{align*}(\sigma_1,\sigma_2)(i)&=\sigma_1(i)\text{ for }i\in\{1,\dots,n\}\\ (\sigma_1,\sigma_2)(i')&=\sigma_2(i)'\text{ for }i\in\{1,\dots,n\}.\end{align*}$$

Now for any representation $\chi$ of $H$ we have an irreducible representation of $W(D_{2n})$ given by $$\mathrm{Ind}_{C_2^{2n-1}\rtimes H}^{C_2^{2n-1}\rtimes S_{2n}}(\epsilon\chi),$$ where $\epsilon\colon C_2^{2n-1}\to \mathbb C^\times$ sends $(z_1,\dots,z_n,z_{1'},\dots,z_{n'})\in C^{2n-1}\subset C^{2n}$ to $z_1\cdots z_n=z_{1'}\cdots z_{n'}$. Tensoring with sign gives $$\mathrm{Ind}_{C_2^{2n-1}\rtimes H}^{C_2^{2n-1}\rtimes S_{2n}}(\epsilon(\chi\otimes\mathrm{sgn})),$$ so it suffices to determine when a representation $\chi$ of $H$ satisfies $\chi\otimes\mathrm{sgn}\simeq\chi$.

There are two interesting cases: let $(V,\pi)$ be an irreducible representation of $S_n$. Then define $(V\otimes V,\chi_\pm^\pi)$, with action defined by: $$ \begin{align*}\chi_\pm(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_\pm((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=\pm\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^\pm}$, when $\pi$ corresponds to the partition $\lambda$ of $n$.

Since $\iota\in H$ has sign $(-1)^n$, we see $-\otimes\mathrm{sgn}$ takes $\chi_\pm^\pi$ to $\chi_\pm^{\pi\otimes\mathrm{sgn}}$ if $n$ is even and to $\chi_\mp^\pi$ if $n$ is odd. Hence $$V^{(\lambda,\lambda)^+}\otimes\mathrm{sgn}\simeq\begin{cases} V^{(\lambda,\lambda)^+}&n\equiv0\pmod2\\ V^{(\lambda,\lambda)^-}&n\equiv1\pmod2. \end{cases}$$