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Kenta Suzuki
Kenta Suzuki
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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 $\chi$ can be one of the following$(V\otimes V,\chi_\pm^\pi)$, 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 $n$.

  • $(V\otimes V,\chi_1^\pi)$, with action defined by: $$ \begin{align*}\chi_1(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_1((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^-}$, when $\pi$ corresponds to the partition $\lambda$ of $n$.
  • $(V\otimes V,\chi_2^\pi)$, with action defined by: $$ \begin{align*}\chi_2(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_2((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=-\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^-}$.

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

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 $\chi$ can be one of the following:

  • $(V\otimes V,\chi_1^\pi)$, with action defined by: $$ \begin{align*}\chi_1(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_1((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^-}$, when $\pi$ corresponds to the partition $\lambda$ of $n$.
  • $(V\otimes V,\chi_2^\pi)$, with action defined by: $$ \begin{align*}\chi_2(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_2((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=-\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^-}$.

From this description, it is clear $-\otimes\mathrm{sgn}$ takes $\chi_i^\pi$ to $\chi_i^{\pi\otimes\mathrm{sgn}}$, and hence $V^{(\lambda,\lambda)+}$ to $V^{(\lambda',\lambda')+}$ (even when $\pi\simeq\pi\otimes\mathrm{sgn}$).

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 $$V^{(\lambda,\lambda)^+}\otimes\mathrm{sgn}\simeq\begin{cases} V^{(\lambda,\lambda)^+}&n\equiv0\pmod2\\ V^{(\lambda,\lambda)^-}&n\equiv1\pmod2. \end{cases}$$

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Kenta Suzuki
Kenta Suzuki
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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 $\chi$ can be one of the following:

  • $(V\otimes V,\chi_1^\pi)$, with action defined by: $$ \begin{align*}\chi_1(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_1((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^-}$, when $\pi$ corresponds to the partition $\lambda$ of $n$.
  • $(V\otimes V,\chi_2^\pi)$, with action defined by: $$ \begin{align*}\chi_2(\sigma_1,\sigma_2)(v_1\otimes v_2)&=\pi(\sigma_1)v_1\otimes\pi(\sigma_2)v_2\\ \chi_2((\sigma_1,\sigma_2)\iota)(v_1\otimes v_2)&=-\pi(\sigma_1)v_2\otimes\pi(\sigma_2)v_1. \end{align*}$$ This corresponds to $V^{(\lambda,\lambda)^-}$.

From this description, it is clear $-\otimes\mathrm{sgn}$ takes $\chi_i^\pi$ to $\chi_i^{\pi\otimes\mathrm{sgn}}$, and hence $V^{(\lambda,\lambda)+}$ to $V^{(\lambda',\lambda')+}$ (even when $\pi\simeq\pi\otimes\mathrm{sgn}$).