Spinorial Representations of Symmetric Groups

TL;DR

This paper investigates when real representations of symmetric and related groups lift to their double covers, providing criteria based on character values and computing associated topological invariants.

Contribution

It introduces new criteria for lifting representations to double covers of orthogonal groups and calculates the second Stiefel-Whitney classes for these cases.

Findings

Criteria for lifting to double covers based on character values

Explicit computation of second Stiefel-Whitney classes for symmetric groups

Application to products of symmetric groups

Abstract

A real representation $\pi$ of a finite group may be regarded as a homomorphism to an orthogonal group $\Or(V)$. For symmetric groups $S_n$, alternating groups $A_n$, and products $S_n \times S_{n'}$ of symmetric groups, we give criteria for whether $\pi$ lifts to the double cover $\Pin(V)$ of $\Or(V)$, in terms of character values. From these criteria, we compute the second Stiefel-Whitney classes of these representations.

Tables

Table 1. Table 1. Spinoriality/Chirality of σ λ subscript 𝜎 𝜆 \sigma_{\lambda} with 2 ≤ | λ | ≤ 6 2 𝜆 6 2\leq|\lambda|\leq 6

$\lambda$	Chirality of ${\sigma }_{\lambda }$	Spinoriality of ${\sigma }_{\lambda }$	Spinoriality of ${{\sigma }_{\lambda }|}_{A_n}$
$\left|\lambda \right|=2$
$\left(2\right)$	achiral	spinorial	spinorial
$\left(1^2\right)$	chiral	aspinorial	spinorial
$\left|\lambda \right|=3$
$\left(3\right)$	achiral	spinorial	spinorial
$(2,1)$	chiral	aspinorial	spinorial
$\left(1^3\right)$	chiral	aspinorial	spinorial
$\left|\lambda \right|=4$
$\left(4\right)$	achiral	spinorial	spinorial
$(3,1)$	chiral	aspinorial	aspinorial
$(2,2)$	chiral	aspinorial	spinorial
$(2,1^2)$	achiral	aspinorial	aspinorial
$\left(1^4\right)$	chiral	aspinorial	spinorial
$\left|\lambda \right|=5$
$\left(5\right)$	achiral	spinorial	spinorial
$(4,1)$	chiral	aspinorial	aspinorial
$(3,2)$	achiral	aspinorial	aspinorial
$(3,1^2)$	chiral	spinorial	spinorial
$(2^2,1)$	chiral	aspinorial	aspinorial
$(2,1^3)$	chiral	aspinorial	aspinorial
$\left(1^5\right)$	chiral	aspinorial	spinorial
$\left|\lambda \right|=6$
$\left(6\right)$	achiral	spinorial	spinorial
$(5,1)$	chiral	aspinorial	aspinorial
$(4,2)$	chiral	aspinorial	spinorial
$(4,1^2)$	achiral	aspinorial	aspinorial
$\left(3^2\right)$	achiral	aspinorial	aspinorial
$(3,2,1)$	achiral	spinorial	spinorial
$(3,1^3)$	achiral	aspinorial	aspinorial
$\left(2^3\right)$	chiral	aspinorial	aspinorial
$(2^2,1^2)$	achiral	aspinorial	spinorial
$(2,1^4)$	achiral	aspinorial	aspinorial
$\left(1^6\right)$	chiral	aspinorial	spinorial

Table 2. Table 2. Spinoriality of σ λ ± superscript subscript 𝜎 𝜆 plus-or-minus \sigma_{\lambda}^{\pm} with λ = λ ′ 𝜆 superscript 𝜆 ′ \lambda=\lambda^{\prime} , ϵ λ = 1 subscript italic-ϵ 𝜆 1 \epsilon_{\lambda}=1 , and 3 ≤ | λ | ≤ 15 3 𝜆 15 3\leq|\lambda|\leq 15

$\lambda$	$\left|\lambda \right|$	${\sigma }_{\lambda }^{\pm }$
$(3,1,1)$	$5$	aspinorial
$(3,2,1)$	$6$	spinorial
$(5,1^4)$	$9$	spinorial
$(5,2,1^3)$	$10$	spinorial
$(4,3,2,1)$	$10$	spinorial
$(4,3,3,1)$	$11$	aspinorial
$(7,1^6)$	$13$	spinorial
$(7,2,1^5)$	$14$	spinorial
$(6,3^2,1^3)$	$15$	spinorial
$(5,4,3,2,1)$	$15$	spinorial
$(4^3,3)$	$15$	spinorial

Equations

$$w_2(\pi )=\left[\frac{{\chi }_{\pi }(1)-{\chi }_{\pi }(s_1)}{4}\right]e_{\mathrm{cup}}+\frac{{\chi }_{\pi }(1)-{\chi }_{\pi }(s_1{s}_3)}{4}{w}_2({\pi }_n),$$

$$(\mathrm{deg}\Pi +1)g{g}^{\prime }\equiv 0\mathrm{mod}2.$$

$$\{v\otimes v+|v{|}^2:v\in V\}.$$

$$C(V)=C(V{)}^0\oplus C(V{)}^1,$$

$$N:C(V)\to C(V)$$

$${\Gamma }_V=\{x\in C(V{)}^{\times }\mid \alpha (x)V{x}^{-1}=V\}.$$

$$c_i^2=(-c_i{)}^2=(-1{)}^{\frac{1}{2}{g}_{\pi }(g_{\pi }+1)},$$

$$\underset{n\to \infty }{\lim }\frac{\#\{\lambda ⊢n\mid {\sigma }_{\lambda }\text{ is achiral and spinorial }\}}{p(n)}=1.$$

$${\chi }_{\lambda }(1)\equiv {\chi }_{\lambda }(s_1)\equiv {\chi }_{\lambda }(s_1{s}_3)\equiv 0\mathrm{mod}8.$$

$${\chi }_{\lambda }(w_{\underline{\mu }})=\sum _{\nu ⊢k}{\chi }_{\nu }(w_{\mu })\cdot f_{\lambda /\nu }.$$

$${\chi }_{\lambda }(s_1)=f_{\lambda /(2)}-f_{\lambda /(1,1)},$$

$${\chi }_{\lambda }(1)=f_{\lambda /(2)}+f_{\lambda /(1,1)},$$

$${\chi }_{\lambda }(s_1{s}_3)=f_{\lambda /(4)}-f_{\lambda /(3,1)}+2f_{\lambda /(2,2)}-f_{\lambda /(2,1,1)}+f_{\lambda /(1,1,1,1)}.$$

$${\chi }_{\lambda }(1)=f_{\lambda /(4)}+3f_{\lambda /(3,1)}+2f_{\lambda /(2,2)}+3f_{\lambda /(2,1,1)}+f_{\lambda /(1,1,1,1)}.$$

$$\left(\genfrac{}{}{0ex}{}{n}{{\lambda }_1,\dots ,{\lambda }_{\ell }}\right).$$

$${\Theta }_{\lambda }(s_1)=\sum _{|{\lambda }_i|\ge 2}\left(\genfrac{}{}{0ex}{}{n-2}{{\lambda }_i,\dots ,{\lambda }_i-2,\dots ,{\lambda }_{\ell }}\right),$$

$$\sum _{\begin{array}{c}1\le i<j\le l\\ |{\lambda }_i|\ge 2,|{\lambda }_j|\ge 2\end{array}}\left(\genfrac{}{}{0ex}{}{n-4}{{\lambda }_1,\dots ,{\lambda }_i-2,\dots ,{\lambda }_j-2,\dots ,{\lambda }_l}\right)+\sum _{|{\lambda }_k|\ge 4}\left(\genfrac{}{}{0ex}{}{n-4}{{\lambda }_1,\dots ,{\lambda }_i-4,\dots ,{\lambda }_l}\right).$$

$$u_i=s_1{s}_{i+1},(i=1,2,\dots ,n-2)$$

$$u_1^3=u_j^2$$

$$(u_i{u}_j{)}^2$$

$$(c_1{c}_2{)}^3=(c_2{c}_3{)}^3=\dots =1.$$

$${\chi }_{\lambda }^{+}(s_1{s}_3)={\chi }_{\lambda }^{-}(s_1{s}_3)=\frac{1}{2}{\chi }_{\lambda }(s_1{s}_3).$$

$$\begin{array}{rl}({\pi }_{\lambda }^{+}{)}_{\mathbb{R}}\oplus ({\pi }_{\lambda }^{-}{)}_{\mathbb{R}}& \cong ({\sigma }_{\lambda }{|}_{A_n}\otimes \mathbb{C}{)}_{\mathbb{R}}\\ & \cong {\sigma }_{\lambda }{|}_{A_n}\oplus {\sigma }_{\lambda }{|}_{A_n},\end{array}$$

$$({\pi }_{\lambda }^{+}{)}_{\mathbb{R}}\cong ({\pi }_{\lambda }^{-}{)}_{\mathbb{R}}\cong {\sigma }_{\lambda }{|}_{A_n}.$$

$$w(\pi )=w_0(\pi )+w_1(\pi )+\cdots +w_d(\pi )\in H^{\ast }(G)=\underset{i=0}{\overset{\infty }{⨁}}{H}^i(G),$$

$$\phi (g)=g\oplus \det (g),$$

$$\tilde{\phi }(x)=\{\begin{array}{ll}& \iota (x),\text{ if x∈Spin(V)}\\ & \iota (x)v^{\prime },\text{ if x∈/Spin(V)}\end{array}$$

$${\rho }^{\prime }\circ \tilde{\phi }=\phi \circ \rho .$$

$$({\rho }^{\prime }{)}^{-1}{\Phi }_V={\tilde{\Phi }}_V.$$

$${\rho }^{\prime }\circ {\hat{\pi }}^{\prime }=\phi \circ \pi ,$$

$$\phi \circ \rho \circ \hat{\pi }=\phi \circ \pi .$$

Full text

1. 1 Introduction
2. 2 Notation and Preliminaries
3. 3 Symmetric Groups
4. 4 Alternating Groups
5. 5 Tables
6. 6 Stiefel-Whitney Classes
7. 7 Products

1. Introduction

A real representation $\pi$ of a finite group $G$ can be viewed as a group homomorphism from $G$ to the orthogonal group $\operatorname{O}(V)$ of a Euclidean space $V$. Recall the double cover $\rho:\operatorname{Pin}(V)\to\operatorname{O}(V)$. We say that $\pi$ is spinorial, provided it lifts to $\operatorname{Pin}(V)$, meaning there is a homomorphism $\hat{\pi}:G\to\operatorname{Pin}(V)$ so that $\rho\circ\hat{\pi}=\pi$.

When the image of $\pi$ lands in $\operatorname{SO}(V)$, the representation is spinorial precisely when its second Stiefel-Whitney class $w_{2}(\pi)$ vanishes. Equivalently, when the associated vector bundle over the classifying space $BG$ has a spin structure. (See Section 2.6 of [Ben98], [GKT89], and Theorem II.1.7 in [LM16].) Determining spinoriality of Galois representations also has applications in number theory: see [Ser84], [Del76], and [PR95].

In this paper we give lifting criteria for representations of the symmetric groups $S_{n}$, the alternating groups $A_{n}$, and a product $S_{n}\times S_{n^{\prime}}$ of two symmetric groups. Write $s_{i}\in S_{n}$ for the transposition $(i,i+1)$, in cycle notation. A key result of this paper is the following:

Theorem 1.1**.**

Let $n\geq 4$.

1. (1)

A representation $\pi$ of $S_{n}$ is spinorial iff $\chi_{\pi}(1)\equiv\chi_{\pi}(s_{1}s_{3})\mod 8$ and $\chi_{\pi}(1)-\chi_{\pi}(s_{1})$ is congruent to [math] or $6\mod 8$. 2. (2)

A representation $\pi$ of $A_{n}$ is spinorial iff $\chi_{\pi}(1)\equiv\chi_{\pi}(s_{1}s_{3})\mod 8$.

Combining this with the main result of [GPS] on character values, one deduces that as $n\to\infty$, ``$100\%$" irreducible representations of $S_{n}$ are spinorial. (See Corollary 3.6.)

Next, we leverage this result to compute the second Stiefel-Whitney classes for (real) representations $\pi$ of $S_{n}$:

[TABLE]

where $\pi_{n}$ is the standard representation of $S_{n}$, and $e_{\mathrm{cup}}\in H^{2}(G,\mathbb{Z}/2\mathbb{Z})$ is a certain cup product.

This formula allows us to compute the second Stiefel-Whitney classes of representations of $S_{n}\times S_{n^{\prime}}$ through Künneth theory, and therefore to identify spinorial representations of this product. To state the result, let $\Pi=\pi\boxtimes\pi^{\prime}$ be the external tensor product of representations $\pi$ of $S_{n}$ and $\pi^{\prime}$ of $S_{n^{\prime}}$. Let $g=\frac{1}{2}(\chi_{\pi}(1)-\chi_{\pi}(s_{1}))$, the multiplicity of $-1$ as an eigenvalue of $\pi(s_{1})$ and similarly write $g^{\prime}$ for the corresponding quantity for $\pi^{\prime}$.

Theorem 1.2**.**

The representation $\Pi$ of $S_{n}\times S_{n^{\prime}}$ is spinorial iff the restrictions of $\Pi$ to $S_{n}\times\{1\}$ and $\{1\}\times S_{n^{\prime}}$ are spinorial, and

[TABLE]

We now describe the layout of this paper. Section 2 reviews the group $\operatorname{Pin}(V)$ and other conventions. The $S_{n}$ case of Theorem 1.1 is proven in Section 3 by means of defining relations for the $s_{i}$. Additionally we note corollaries of Theorem 1.1: primarily the aforementioned ``$100\%$" result, a connection with skew Young tableau numbers, and the important case of permutation modules, meaning the induction of the trivial character from a Young subgroup of $S_{n}$. In particular we demonstrate that the regular representation of $S_{n}$ is spinorial for $n\geq 4$.

Representations of the alternating groups are treated in Section 4, again via generators and relations. The main result is the $A_{n}$ case of Theorem 1.1. We enumerate the spinorial irreducible representations of $A_{n}$ in Theorem 4.2. Data for spinoriality of irreducible representations of $S_{n}$ and $A_{n}$ for small $n$ is presented in Tables 1 and 2 of Section 5.

In Section 6 we review the axioms of Stiefel-Whitney classes of real representations, and then deduce the Stiefel-Whitney class of a real representation of $S_{n}$. In Section 7 we apply Künneth theory to this formula to compute Stiefel-Whitney classes for real representations of $S_{n}\times S_{n^{\prime}}$. From this it is straightforward to deduce Theorem 1.2.

Acknowledgements

The authors would like to thank Dipendra Prasad, Rohit Joshi, and Amritanshu Prasad for helpful conversations. The research by the first author for this article was supported by a PhD scholarship from the Indian National Board for Higher Mathematics.

2. Notation and Preliminaries

2.1. Representations

All representations are on finite-dimensional vector spaces, which are always real, except in Section 4, where they may be specified as complex. For a representation $(\pi,V)$ of a group $G$, write $\det\pi$' for the composition $\det\circ\pi$; it is a linear character of $G$. Also write $\chi_{\pi}$' for the character of $\pi$. If $H\leq G$ is a subgroup, write $\pi|_{H}$ for the restriction of $\pi$ to $H$. A real representation $\pi:G\to\operatorname{GL}(V)$ can be conjugated to have image in $\operatorname{O}(V)$, so we will assume that this is the case. When $\det\pi$ is trivial, it maps to $\operatorname{SO}(V)$, and the spinoriality question is whether it lifts to the double cover $\operatorname{Spin}(V)$ (which we review in the next section).

Let $\operatorname{sgn}:S_{n}\to\{\pm 1\}$ be the usual sign character. For $G=S_{n}$, we say that $\pi$ is chiral provided $\det\pi=\operatorname{sgn}$ and $\pi$ is achiral provided $\det\pi=1$. Write $\pi_{n}:S_{n}\to\operatorname{GL}_{n}(\mathbb{R})$ for the standard representation of $S_{n}$ by permutation matrices.

2.2. The Pin Group

We essentially review [BtD95, Chapter $1.6$] for defining the groups $\operatorname{Spin}(V)$ and $\operatorname{Pin}(V)$, where $V$ is a Euclidean (i.e., a normed finite-dimensional real vector) space. The Clifford algebra $C(V)$ is the quotient of the tensor algebra $T(V)$ by the two-sided ideal generated by the set

[TABLE]

Write $C(V)^{\times}$ for its group of units.

We identify $V$ as a subspace of $C(V)$ through the natural injection $i:V\to C(V)$. Write $\alpha$ for the unique involution of the $\mathbb{R}$-algebra $C(V)$ with the property that $\alpha(x)=-x$ for $x\in V$. One has

[TABLE]

where $C(V)^{0}$ is the $1$-eigenspace of $\alpha$ and $C(V)^{1}$ is the $-1$-eigenspace.

Write $t$ for the unique anti-involution of $C(V)$ with $t(x)=x$. For $x\in C(V)$, define $\overline{x}=t(\alpha(x))$; it is again an algebra anti-involution. Define

[TABLE]

by $N(x)=x\overline{x}$. Put

[TABLE]

Let $\rho:\Gamma_{V}\to\operatorname{GL}(V)$ be the homomorphism given by $v\mapsto\alpha(x)vx^{-1}$. We will repeatedly use the fact that if $v$ is a unit vector, then $\rho(v)$ is the reflection determined by $v$. Write $\operatorname{Pin}(V)$' for the kernel of the restriction of $N$ to $\Gamma_{V}$. The restriction of $\rho$ to $\operatorname{Pin}(V)$ is a double cover of $\operatorname{O}(V)$ with kernel $\{\pm 1\}$. The preimage of $\operatorname{SO}(V)$ under $\rho$ is denoted $\operatorname{Spin}(V)$'. Alternately, $\operatorname{Spin}(V)=\operatorname{Pin}(V)\cap C(V)^{0}$.

3. Symmetric Groups

3.1. Lifting Criteria

Let $n\geq 2$. The group $S_{n}$ is generated by the transpositions $s_{i}=(i,i+1)$ for $1\leq i\leq n-1$, with the following relations:

1. (1)

${s_{i}}^{2}=1,\quad 1\leq i\leq n-1$, 2. (2)

$s_{i}s_{k}=s_{k}s_{i},\text{ when }\quad|i-k|>1$, 3. (3)

${(s_{i}s_{i+1})}^{3}=1,\quad 1\leq i\leq n-2$.

Therefore, defining a homomorphism from $S_{n}$ to a group $G$ is equivalent to choosing elements $x_{1},\ldots,x_{n-1}\in G$ satisfying the same relations. Let us call the relation $x_{i}^{2}=1$ the first lifting condition", the relation $x_{i}x_{k}=x_{k}x_{i}$ the second lifting condition", and $(x_{i}x_{i+1})^{3}=1$ the ``third lifting condition". Note that this second condition is vacuous for $n<4$.

Let $\pi:S_{n}\to\operatorname{O}(V)$ be a representation of degree $d$. For each $\pi(s_{i})\in\operatorname{O}(V)$ there are $\pm c_{i}\in\operatorname{Pin}(V)$ with $\rho(\pm c_{i})=\pi(s_{i})$, and the question is whether we may choose signs so that the $x_{i}=\pm c_{i}$ satisfy these lifting conditions.

Let $g_{\pi}=\dfrac{\chi_{\pi}(1)-\chi_{\pi}(s_{1})}{2}$, as in [APS17]. This is the multiplicity of the eigenvalue $-1$ of $\pi(s_{1})$, and the eigenvalue $1$ occurs with multiplicity $d-g_{\pi}$. Put $c_{i}=u_{1}\cdots u_{g_{\pi}}\in\operatorname{Pin}(V)$, where $u_{1},\ldots,u_{g_{\pi}}$ is an orthonormal basis of the $-1$-eigenspace of $\pi(s_{i})$. Since $\pi(s_{i})$ is the product of the reflections in each $u_{j}$, the elements $c_{i}$ and $-c_{i}$ are the lifts of $\pi(s_{i})$. One computes that

[TABLE]

and therefore the first lifting condition is satisfied iff $g_{\pi}$ is congruent to [math] or $3$ modulo $4$. It does not matter for this whether we choose $c_{i}$ or $-c_{i}$.

Consider the sequence $(c_{1}c_{2})^{3},(c_{2}c_{3})^{3},\ldots\in\operatorname{Pin}(V)$. Since each $(\pi(s_{i})\pi(s_{i+1}))^{3}=1$, this must be a sequence of $\pm 1$'s. For the third lifting condition these must each be $1$. Thus $c_{1}$ may take either sign, but then the signs for $c_{2},c_{3},\ldots$ are determined. Moreover this does not affect the first lifting condition. Thus:

Proposition 3.1**.**

The first and third lifting conditions hold iff $g_{\pi}\equiv 0$ or $3\mod 4$.

Now let $|i-k|>1$, and suppose as above that $c_{i}^{2}=1=c_{k}^{2}=1$. Then the second lifting condition holds iff $(c_{i}c_{k})^{2}=1$. By conjugating we may assume that $i=1$ and $k=3$. So put $h_{\pi}=\dfrac{\chi_{\pi}(1)-\chi_{\pi}(s_{1}s_{3})}{2}$; as above the condition is equivalent to $h_{\pi}\equiv 0,3\mod 4$. However:

Lemma 3.2**.**

The integer $h_{\pi}$ is even.

Proof.

Let $\zeta_{4}$ be a $4$-cycle in $S_{n}$. Then $\zeta_{4}^{2}$ is conjugate to $s_{1}s_{3}$. Let $m$ be the multiplicity of $i=\sqrt{-1}$ as an eigenvalue of $\pi(\zeta_{4})$. Then $h_{\pi}$, the multiplicity of $-1$ as an eigenvalue of $\pi(s_{1}s_{3})$, is $2m$. ∎

Therefore:

Proposition 3.3**.**

The second lifting condition holds iff $h_{\pi}$ is a multiple of $4$.

∎

We summarize the above as the following:

Theorem 3.4**.**

Let $n\geq 4$, and $\pi$ a representation of $S_{n}$. The following are equivalent:

1. (1)

The representation $\pi$ is spinorial. 2. (2)

$g_{\pi}\equiv 0$* or $3\mod 4$, and $h_{\pi}\equiv 0\mod 4$.* 3. (3)

$\chi_{\pi}(1)-\chi_{\pi}(s_{1})\equiv 0$* or $6\mod 8$, and $\chi_{\pi}(1)\equiv\chi_{\pi}(s_{1}s_{3})\mod 8$.*

When $\pi$ is spinorial it has two lifts.

When $\pi$ is spinorial, note that $g_{\pi}\equiv 0$ iff $\pi$ is achiral, and $g_{\pi}\equiv 3$ iff $\pi$ is chiral.

Proof.

The equivalence should be clear; the two lifts correspond to the choice of sign for $c_{1}$. ∎

Remark: The two lifts correspond to the two members of $H^{1}(S_{n},\mathbb{Z}/2\mathbb{Z})$; see Theorem II.1.7 of [LM16].

Corollary 3.5**.**

A representation $\pi$ of $S_{n}$ is spinorial iff its restrictions to the cyclic subgroups $\langle s_{1}\rangle$ and $\langle s_{1}s_{3}\rangle$ are both spinorial.

Proof.

Indeed, the $g_{\pi}$ condition corresponds to the subgroup $\langle s_{1}\rangle$, and the $h_{\pi}$ condition corresponds to $\langle s_{1}s_{3}\rangle$. ∎

The irreducible representations of $S_{n}$ are the Specht modules $(\sigma_{\lambda},V_{\lambda})$, indexed by partitions of $n$. (See for instance [JK09].) Write $f_{\lambda}=f_{\pi_{\lambda}}$, and similarly for $g_{\lambda},h_{\lambda}$. Write $p(n)$ for the number of partitions of $n$.

Corollary 3.6**.**

We have

[TABLE]

In other words, as $n\to\infty$, $100\%$ of irreducible representations of $S_{n}$ are achiral and spinorial.

Proof.

According to [GPS], as $n\to\infty$, $100\%$ of partitions $\lambda$ of $n$ have

[TABLE]

The conclusion then follows from Theorem 1.1.

∎

3.2. Connection with Skew Young Tableaux

Let $\mu,\lambda$ be partitions for which the Young diagram of $\lambda$ contains that of $\mu$. The notion of standard Young tableaux generalizes to ``skew diagrams" $\lambda/\mu$. Following Section 7.10 in [Sta99], write $f_{\lambda/\mu}$ for the number of SYT on $\lambda/\mu$. (If the Young diagram of $\mu$ is not contained in that of $\lambda$, put $f_{\lambda/\mu}=0$.)

Proposition 3.7**.**

We have

1. (1)

$g_{\lambda}=f_{\lambda/(1,1)}$* and* 2. (2)

$h_{\lambda}=2\cdot(f_{\lambda/(3,1)}+f_{\lambda/(2,1,1)})$.

Proof.

Let $\mu\vdash k$ for some $k\leq n$, and let $\underline{\mu}$ be the partition of $n$ defined by adding $(n-k)$ $1$'s, i.e. $\underline{\mu}=\mu+\underbrace{1+\cdots+1}_{(n-k)\text{ times}}$. Write $w_{\mu}\in S_{n}$ be a permutation with cycle type $\mu$. According to [Sta99], Exercise 7.62, we have

[TABLE]

Taking $\mu=(2)$ gives

[TABLE]

and taking $\mu=(1,1)$ gives

[TABLE]

so that $g_{\lambda}=f_{\lambda/(1,1)}$.

Similarly, taking $\mu=(2,2)$ and using the character table for $S_{4}$, we compute

[TABLE]

Taking $\mu=(1,1,1,1)$ gives

[TABLE]

Combining (3.1) and (3.2) gives the formula for $h_{\lambda}$.

∎

3.3. Permutation Representations

Another important class of representations of $S_{n}$ are the permutation representations, which are also indexed by partitions of $n$. Let $\lambda=(\lambda_{1},\ldots,\lambda_{\ell})\vdash n$, and consider the set $\mathcal{P}_{\lambda}$ of ordered partitions of $\{1,2,\ldots,n\}$ with shape $\lambda$. Thus a member of $\mathcal{P}_{\lambda}$ is an $\ell$-tuple $(X_{1},\ldots,X_{\ell})$ of disjoint sets with each $|X_{i}|=\lambda_{i}$ and $\bigcup_{i=1}^{\ell}X_{i}=\{1,\ldots,n\}$. Note that $\mathcal{P}_{\lambda}$ has cardinality

[TABLE]

The group $S_{n}$ acts on $\mathcal{P}_{\lambda}$ in the obvious way, and we obtain the permutation representation $\mathbb{R}[\mathcal{P}_{\lambda}]$. This representation space is given by formal linear combinations of elements of $\mathcal{P}_{\lambda}$, so its degree is given by (3.3).

For example, if $\lambda=(1,\ldots,1)\vdash n$, then $\mathbb{R}[\mathcal{P}_{\lambda}]$ is the regular representation of $S_{n}$. If $\lambda=(n-1,1)$, then $\mathbb{R}[\mathcal{P}_{\lambda}]$ is the standard representation $\pi_{n}$ of $S_{n}$ on $\mathbb{R}^{n}$. Note that $S_{n}$ acts transitively on $\mathcal{P}_{\lambda}$ with a stabilizer equal to the ``Young subgroup" $S_{\lambda}=S_{\lambda_{1}}\times\cdots\times S_{\lambda_{\ell}}$, so we can also view $\mathbb{R}[\mathcal{P}_{\lambda}]$ as the induction from $S_{\lambda}$ to $S_{n}$ of the trivial representation.

The characters of the $\mathbb{R}[\mathcal{P}_{\lambda}]$, though typically reducible, form an important basis of the representation ring of $S_{n}$. See, for example, Section 2.2 of [JK09].

Recall that, for a representation $\pi$ coming from a group action, the character value $\chi_{\pi}(g)$ is the number of fixed points of $g$. Write $\Theta_{\lambda}$ for the character $\chi_{\mathbb{R}[\mathcal{P}_{\lambda}]}$.

From this fixed-point principle we compute

[TABLE]

since a partition in $\mathcal{P}_{\lambda}$ is fixed by $s_{1}$ iff $1$ and $2$ lie in the same part $X_{i}$ for some $i$. Similarly, $\Theta_{\lambda}(s_{1}s_{3})$ equals

[TABLE]

This is because a partition in $\mathcal{P}_{\lambda}$ is fixed by $s_{1}s_{3}$ iff either the elements $1,2,3,4$ all lie in the same part $X_{i}$, or $1,2$ lie in some $X_{i}$ and $3,4$ lie some other part $X_{j}$.

These character values may be used to compute $g_{\mathbb{R}[\mathcal{P}_{\lambda}]}$ and $h_{\mathbb{R}[\mathcal{P}_{\lambda}]}$. (Compare Lemma 17 in [APS17].) For instance if $\lambda=(1,\ldots,1)\vdash n$, then $\Theta_{\lambda}(s_{1})=\Theta_{\lambda}(s_{1}s_{3})=0$, so $g_{\mathbb{R}[\mathcal{P}_{\lambda}]}=h_{\mathbb{R}[\mathcal{P}_{\lambda}]}=\frac{n!}{2}$. Thus the regular representation of $S_{n}$ is achiral and spinorial.

The standard representation $\pi_{n}$ corresponds to $\lambda=(n-1,1)$. For this $\lambda$ we have $\Theta_{\lambda}(s_{1})=n-2$, so $g_{\mathbb{R}[\mathcal{P}_{\lambda}]}=1$ and it follows that $\pi_{n}$ is aspinorial.

For easy reference, we collect here results about common representations of $S_{n}$:

Proposition 3.8**.**

For $n\geq 2$ the standard representation $\pi_{n}$ is achiral and aspinorial, and the sign representation is achiral and aspinorial. For $n\geq 4$, the regular representation of $S_{n}$ is achiral and spinorial.

4. Alternating Groups

Now we turn to the alternating group $A_{n}$, for $n\geq 4$.

4.1. Spinoriality Criterion

The group $A_{n}$ is generated by the permutations

[TABLE]

with relations:

[TABLE]

(See for instance [CM80].)

Note that $u_{1}$ is a $3$-cycle and the other $u_{i}$ are $(2,2)$-cycles.

For a real representation $(\pi,V)$ of $A_{n}$ again put $h_{\pi}=\dfrac{\chi_{\pi}(1)-\chi_{\pi}(s_{1}s_{3})}{2}$. Since this is the multiplicity of the eigenvalue $-1$ of $\pi(s_{1}s_{3})$, which has determinant $1$, the integer $h_{\pi}$ is necessarily even.

Theorem 4.1**.**

A real representation $(\pi,V)$ of $A_{n}$ is spinorial if and only if $h_{\pi}$ is a multiple of $4$. In this case there is a unique lift.

Proof.

As in Section 3.1 we much choose $c_{i}$ with $\rho(c_{i})=\pi(u_{i})$ satisfying the same relations as the $u_{i}$. Let $c_{1}$ be a lift of $\pi(u_{1})$. Since $\rho(c_{1})^{3}=\pi(u_{1})^{3}=1$, we have $c_{1}^{3}=\pm 1$. This determines the sign of $c_{1}$.

The $u_{j}$ and $u_{i}u_{j}$ as above, for $j>1$, are all conjugate to $u_{2}$ in $A_{n}$. Therefore all the conditions $c_{j}^{2}=1$ and $(c_{i}c_{j})^{2}=1$ are equivalent to the condition $c_{2}^{2}=1$. As before, this is equivalent to $h_{\pi}$ being congruent to [math] or $3\mod 4$, but since $h_{\pi}$ is even, it must be a multiple of $4$.

Finally, there is a unique choice of signs normalizing $c_{2},\ldots,c_{n-2}$ so that

[TABLE]

∎

Example: If $\rho$ is the regular representation of $A_{n}$ (on the group algebra $\mathbb{R}[A_{n}]$), then $h_{\rho}=\dfrac{n!}{4}$, so $\rho$ is spinorial iff $n\neq 4,5$.

Example: For the standard representation $\pi_{n}$ of $S_{n}$, $h_{\pi_{n}}=2$, so the restriction of $\pi_{n}$ to $A_{n}$ is aspinorial.

4.2. Real Irreducible Representations

Let us review the relationship between real and complex irreducible representations of a finite group $G$, following [BtD95]. If $(\pi,V)$ is a complex representation of a group $G$, write $(\pi_{\mathbb{R}},V_{\mathbb{R}})$ for the realization of $\pi$, meaning that we simply forget the complex structure on $V$ and regard it as a real representation. If moreover $(\pi,V)$ is an orthogonal complex representation, meaning that it admits a $G$-invariant symmetric nondegenerate bilinear form, then there is a unique real representation $(\pi_{0},V_{0})$, up to isomorphism, so that $\pi\cong\pi_{0}\otimes_{\mathbb{R}}\mathbb{C}$.

It is not hard to see that $\pi_{0}$ is self-dual iff $\pi$ is self-dual, and that an orthogonal $\pi$ is spinorial, i.e., lifts to $\operatorname{Spin}(V)$, iff $\pi_{0}$ is spinorial, i.e., lifts to $\operatorname{Spin}(V_{0})$.

Every real irreducible representation $\sigma$ of $G$ is either of the form

1. (1)

$\sigma=\pi_{0}$, for an orthogonal irreducible complex representation $\pi$ of $G$, or 2. (2)

$\sigma=\pi_{\mathbb{R}}$, for an irreducible complex representation $\pi$ of $G$ which is not orthogonal.

In the case of $G=S_{n}$, all complex representations are orthogonal.

4.3. Real Irreducible Representations of $A_{n}$

For a partition $\lambda$, write $\lambda^{\prime}$ for its conjugate partition. Furthermore write $\epsilon_{\lambda}=1$ when the number of cells in the Young diagram of $\lambda$ above the diagonal is even, and $\epsilon_{\lambda}=-1$ when this number is odd.

For example, let $\lambda=(4,3,2,1)$. Then $\lambda=\lambda^{\prime}$, and there are $4$ cells above the diagonal, shaded in the Young diagram below, so $\epsilon_{\lambda}=1$.

{ytableau}

*(white) & *(gray) *(gray) *(gray)

*(white) *(white) *(gray)

*(white) *(white)

*(white)

Let $\sigma_{\lambda}$ be the (real) Specht module corresponding to $\lambda$, as before. Write $\pi_{\lambda}=\sigma_{\lambda}\otimes\mathbb{C}$ for its complexification, i.e., the complex Specht module corresponding to $\lambda$.

If $\lambda\neq\lambda^{\prime}$, then $\pi_{\lambda}$ restricts irreducibly to $A_{n}$. When $\lambda=\lambda^{\prime}$, the restriction of $\pi_{\lambda}$ to $A_{n}$ decomposes into a direct sum of two nonisomorphic representations $\pi_{\lambda}^{+}$ and $\pi_{\lambda}^{-}$. Either of $\pi_{\lambda}^{\pm}$ is the twist of the other by $\sigma_{\lambda}(w)$ for any odd permutation $w$. The set of $\pi_{\lambda}$ with $\lambda\neq\lambda^{\prime}$, together with the $\pi_{\lambda}^{\pm}$ for $\lambda=\lambda^{\prime}$, is a complete set of irreducible complex representations of $A_{n}$.

For $\lambda=\lambda^{\prime}$ we have

[TABLE]

If moreover $\epsilon_{\lambda}=1$, then the representations $\pi_{\lambda}^{+}$ and $\pi_{\lambda}^{-}$ are orthogonal. We may then define real irreducible representations of $A_{n}$ by $\sigma_{\lambda}^{\pm}=(\pi_{\lambda}^{\pm})_{0}$. However when $\epsilon_{\lambda}=-1$, the representations $\pi_{\lambda}^{\pm}$ are not orthogonal, and therefore the realizations $(\pi_{\lambda}^{\pm})_{\mathbb{R}}$ are irreducible. Since

[TABLE]

we have isomorphisms of real $A_{n}$-representations:

[TABLE]

From these considerations and Theorem 4.1 we conclude:

Theorem 4.2**.**

A complete list of real irreducible representations of $A_{n}$ is given by

1. (1)

$\sigma_{\lambda}|_{A_{n}}$, where either $\lambda\neq\lambda^{\prime}$, or $\lambda=\lambda^{\prime}$ and $\epsilon_{\lambda}=-1$, and 2. (2)

$\sigma_{\lambda}^{\pm}$, where $\lambda=\lambda^{\prime}$ and $\epsilon_{\lambda}=1$.

In the first case, $\sigma_{\lambda}|_{A_{n}}$ is spinorial iff $\chi_{\lambda}(s_{1}s_{3})\equiv\chi_{\lambda}(1)\mod 8$. In the second case, $\sigma_{\lambda}^{+}$ is spinorial iff $\sigma_{\lambda}^{-}$ is spinorial iff $\chi_{\lambda}(s_{1}s_{3})\equiv\chi_{\lambda}(1)\mod 16$.

Remark: When $\lambda=\lambda^{\prime}$ the restriction $\sigma_{\lambda}|_{A_{n}}$ is necessarily spinorial by (4.1), since all $h_{\pi}$ are even.

5. Tables

We illustrate the theory of this paper by means of two tables. Table 1 below contains the following information for $2\leq n\leq 6$:

1. (1)

Whether the Specht Module $\sigma_{\lambda}$ is chiral, i.e., whether $g_{\lambda}$ is odd. 2. (2)

Whether $\sigma_{\lambda}$ is spinorial, by Theorem 3.4. 3. (3)

Whether the restriction of $\sigma_{\lambda}$ to $A_{n}$ is spinorial, by Theorem 4.2.

Table 2 below lists for self-conjugate $\lambda$ with $\epsilon_{\lambda}=1$, whether the constituents $\sigma_{\lambda}^{+}$ and $\sigma_{\lambda}^{-}$ are spinorial, following Theorem 4.2. This is done for all such $\lambda$ with $3\leq|\lambda|\leq 15$.

6. Stiefel-Whitney Classes

6.1. Basic Properties

Let $G$ be a finite group and $\pi$ a real representation of $G$. Stiefel-Whitney classes $w_{i}(\pi)$ are defined for $0\leq i\leq\deg\pi$ as members of the cohomology groups $H^{i}(G)=H^{i}(G,\mathbb{Z}/2\mathbb{Z})$. Here $\mathbb{Z}/2\mathbb{Z}$ is trivial as a $G$-module. One considers the total Stiefel-Whitney class in the $\mathbb{Z}/2\mathbb{Z}$-cohomology ring:

[TABLE]

where $d=\deg\pi$.

According to, for example [GKT89], these characteristic classes satisfy the following properties:

1. (1)

$w_{0}(\pi)=1$. 2. (2)

$w_{1}(\pi)=\det\pi$, regarded as a linear character in $H^{1}(G)\cong\operatorname{Hom}(G,\{\pm 1\})$. 3. (3)

If $\pi^{\prime}$ is another real representation, then $w(\pi\oplus\pi^{\prime})=w(\pi)\cup w(\pi^{\prime})$. 4. (4)

If $f:G^{\prime}\to G$ is a group homomorphism, then $w(\pi\circ f)=f^{*}(w(\pi))$, where $f^{*}$ is the induced map on cohomology. 5. (5)

Suppose $\det\pi=1$. Then $w_{2}(\pi)=0$ iff $\pi$ is spinorial.

Note in particular that $w(\chi)=1+\chi$, if $\chi$ is a linear character of $G$.

The last property generalizes as follows:

Proposition 6.1**.**

A real representation $\pi$ is spinorial iff $w_{2}(\pi)=w_{1}(\pi)\cup w_{1}(\pi)$.

We will deduce this proposition from the following lemma.

Lemma 6.2**.**

Let $\pi^{\prime}=\pi\oplus\det\pi$. Then $\pi$ is spinorial iff $\pi^{\prime}$ is spinorial.

Proof.

Let $V^{\prime}$ be the representation space of $\pi^{\prime}$; say $V^{\prime}=V\oplus\mathbb{R}v^{\prime}$ for some unit vector $v^{\prime}$ perpendicular to $V$. Write $\iota:C(V)\to C(V^{\prime})$ for the canonical injection. Note that if $x\in C(V)^{0}$, then $\iota(x)v^{\prime}=v^{\prime}\cdot\iota(x)$, and if $x\in C(V)^{1}$, then $\iota(x)v^{\prime}=-v^{\prime}\cdot\iota(x)$. Write $\rho^{\prime}:\operatorname{Pin}(V^{\prime})\to O(V^{\prime})$ for the usual double cover.

Define $\varphi:\operatorname{O}(V)\to\operatorname{SO}(V^{\prime})$ by

[TABLE]

so that $\pi^{\prime}=\varphi\circ\pi$. Write $\Phi_{V}<\operatorname{SO}(V^{\prime})$ for the image of $\varphi$. The essential problem is to construct a lift of $\varphi\circ\rho$. Since $\operatorname{Spin}(V)=\operatorname{Pin}(V)\cap C(V)^{0}$, the map $\tilde{\varphi}:\operatorname{Pin}(V)\to\operatorname{Spin}(V^{\prime})$ defined by

[TABLE]

is a group homomorphism. Note that

[TABLE]

Let us see that $\tilde{\varphi}$ is injective; suppose $\tilde{\varphi}(x_{1})=\tilde{\varphi}(x_{2})$. Clearly if $x_{1}$ and $x_{2}$ are both in $\operatorname{Spin}(V)$, or both not in $\operatorname{Spin}(V)$, then $x_{1}=x_{2}$. Suppose $x_{1}\in\operatorname{Spin}(V)$ but $x_{2}\notin\operatorname{Spin}(V)$. Then $v^{\prime}=\iota(x_{2}^{-1}x_{1})$ and in particular $v^{\prime}\in\iota(C(V))$. But this is impossible, say by Corollary 6.7 in Chapter I of [BtD95]. We conclude that $\tilde{\varphi}$ is injective.

Write $\tilde{\Phi}_{V}<\operatorname{Spin}(V^{\prime})$ for the image of $\tilde{\varphi}$; then $\tilde{\varphi}$ is an isomorphism from $\operatorname{Pin}(V)$ onto $\tilde{\Phi}_{V}$ and

[TABLE]

If $\hat{\pi}$ is a lift of $\pi$, then $\tilde{\varphi}\circ\hat{\pi}$ is a lift of $\pi^{\prime}$. Conversely, suppose $\hat{\pi}^{\prime}$ is a lift of $\pi^{\prime}$. Then its image lies in $\tilde{\Phi}_{V}$, and therefore $\hat{\pi}^{\prime}=\tilde{\varphi}\circ\hat{\pi}$ for some homomorphism $\hat{\pi}:G\to\operatorname{Pin}(V)$. Since

[TABLE]

it follows that

[TABLE]

Thus $\hat{\pi}$ is a lift of $\pi$.

∎

Proof.

(of Proposition 6.1) Note that $\det\pi^{\prime}=1$, so $\pi$ is spinorial iff $w_{2}(\pi^{\prime})=0$. But

[TABLE]

whence the theorem. ∎

6.2. The group of order $2$

Let $C$ be a cyclic group of order $2$, and write $\operatorname{sgn}$' for its nontrivial linear character. Then $H^{2}(C)\cong\mathbb{Z}/2\mathbb{Z}$; the nonzero element is $\operatorname{sgn}\cup\operatorname{sgn}$'. Let $\pi$ be the sum of $m$ copies of the trivial representation with $n$ copies of $\operatorname{sgn}$. Then

[TABLE]

In particular, $w_{2}(\pi)=\binom{n}{2}\cdot\operatorname{sgn}\cup\operatorname{sgn}$. By Proposition 6.1, $\pi$ is spinorial iff $n^{2}\equiv\binom{n}{2}\mod 2$; equivalently, $n\equiv 0$ or $3\mod 4$.

6.3. Calculation for $S_{n}$

Write

[TABLE]

Again write $\pi_{n}$ for the standard representation of $S_{n}$ on $\mathbb{R}^{n}$. From [Ser84, Section $1.5$] we know that $e_{\mathrm{cup}}$ and $w_{2}(\pi_{n})$ comprise a basis for the $\mathbb{Z}/2\mathbb{Z}$-vector space $H^{2}(S_{n})$.

Proposition 6.3**.**

The map

[TABLE]

given by the two restrictions, is an isomorphism for $n\geq 4$.

Thus the second $\mathbb{Z}/2\mathbb{Z}$-cohomology of $S_{n}$ is ``detected" by these cyclic subgroups; compare Corollary 3.5 above and Theorem VI.1.2 in [AM04].

Proof.

Since $\Phi$ is a linear map between $2$-dimensional $\mathbb{Z}/2\mathbb{Z}$-vector spaces, it suffices to prove that its rank is $2$. Let $b_{1}$ be the generator of $H^{2}(\langle s_{1}\rangle)$, and $b_{2}$ be the generator of $H^{2}(\langle s_{1}s_{3}\rangle)$.

The restriction of $\pi_{n}$ to $\langle s_{1}\rangle$ decomposes into a trivial $(n-1)$-dimensional representation plus one copy of $\operatorname{sgn}$. The restriction to $\langle s_{1}s_{3}\rangle$ contains two copies of $\operatorname{sgn}$. Therefore $\Phi(w_{2}(\pi_{n}))=(0,b_{2})$. Similarly $\Phi(w_{2}(\operatorname{sgn}\oplus\operatorname{sgn}))=\Phi(e_{\mathrm{cup}})=(b_{1},0)$. Thus $\Phi$ has rank $2$, as required.

∎

Theorem 6.4**.**

For $\pi$ a real representation of $S_{n}$, with $n\geq 4$, we have

[TABLE]

Here $[\cdot]$ denotes the greatest integer function.

Proof.

Suppose first that $\pi$ is achiral. Since $e_{\mathrm{cup}}$ and $w_{2}(\pi_{n})$ form a basis of $H^{2}(S_{n})$ we must have

[TABLE]

for some $c_{1},c_{2}\in\mathbb{Z}/2\mathbb{Z}$. Thus $\Phi(w_{2}(\pi))=c_{1}b_{1}+c_{2}b_{2}$. By the Stiefel-Whitney class properties (4) and (5),

[TABLE]

and

[TABLE]

Thus $c_{1}\equiv\frac{g_{\pi}}{2}\mod 2$ and $c_{2}\equiv\frac{h_{\pi}}{2}\mod 2$.

If $\pi$ is chiral, then $\pi^{\prime}=\pi\oplus\operatorname{sgn}$ is achiral. From the identity $w_{2}(\pi)=w_{2}(\pi^{\prime})+e_{\mathrm{cup}}$, we deduce that

[TABLE]

∎

Remark: For $n=2,3$, similar reasoning gives $H^{2}(S_{n})\cong H^{2}(\langle s_{1}\rangle)$ and

[TABLE]

Remark: Since the groups $H^{2}(A_{n},\mathbb{Z}/2\mathbb{Z})$ have order $1$ or $2$, computing the Stiefel-Whitney class of a real representation of $A_{n}$ is equivalent to determining its spinoriality, which we have already done.

7. Products

Spinoriality for representations of $S_{n}\times S_{n^{\prime}}$ can also be determined by means of generators and relations. (See Theorem 5.4.1 in [Gan19].) However we will instead obtain a satisfactory criterion by simply feeding our calculation of $w_{2}(\pi)$ into the machinery of Stiefel-Whitney classes.

7.1. External tensor products

Let $G,G^{\prime}$ be finite groups, let $(\pi,V)$ be a real representation of $G$, and let $(\pi^{\prime},V^{\prime})$ be a real representation of $G^{\prime}$. Write $\pi\boxtimes\pi^{\prime}$ for the external tensor product representation of $G\times G^{\prime}$ on $V\otimes V^{\prime}$. One computes

[TABLE]

and hence

[TABLE]

which is an element of

[TABLE]

The famous ``splitting principle" (e.g., proceeding as in Problem 7-C of [MS16]) similarly gives

[TABLE]

as an element of

[TABLE]

Finally, $w_{2}(\pi\boxtimes\pi^{\prime})+w_{1}(\pi\boxtimes\pi^{\prime})\cup w_{1}(\pi\boxtimes\pi^{\prime})$ comes out to be

[TABLE]

Thus $\pi\boxtimes\pi^{\prime}$ is spinorial (by Proposition 6.1) iff all of the following vanish:

1. (1)

$\deg\pi^{\prime}\cdot w_{2}(\pi)+\binom{\deg\pi^{\prime}+1}{2}w_{1}(\pi)\cup w_{1}(\pi)$, 2. (2)

$(\deg\pi\deg\pi^{\prime}+1)w_{1}(\pi)\otimes w_{1}(\pi^{\prime})$, and 3. (3)

$\binom{\deg\pi+1}{2}w_{1}(\pi^{\prime})\cup w_{1}(\pi^{\prime})+\deg\pi\cdot w_{2}(\pi^{\prime})$.

7.2. Products of Symmetric Groups

We now prove Theorem 1.2. Let $\pi,\pi^{\prime}$ be representations of $S_{n}$ and $S_{n^{\prime}}$. Write $f=f_{\pi}$, $f^{\prime}=f_{\pi^{\prime}}$ and similarly for $g,h,g^{\prime}$ and $h^{\prime}$. Let $\Pi=\pi\boxtimes\pi^{\prime}$; all representations of $S_{n}\times S_{n^{\prime}}$ are sums of such representations.

Proof.

By Proposition 6.1, $\Pi$ is spinorial iff

[TABLE]

From Theorem 6.4 and (1)-(3) of Section 7.1 we deduce that $\Pi$ is spinorial iff all of the following are even:

1. (1)

$f^{\prime}\cdot\frac{h}{2}$, 2. (2)

$f^{\prime}\left[\dfrac{g}{2}\right]+\binom{f^{\prime}+1}{2}g$, 3. (3)

$(ff^{\prime}+1)gg^{\prime}$, 4. (4)

$f\cdot\frac{h^{\prime}}{2}$, and 5. (5)

$f\left[\dfrac{g^{\prime}}{2}\right]+\binom{f+1}{2}g^{\prime}$.

Note that if $\Pi$ is spinorial, then its restriction to $S_{n}\times\{1\}$, which amounts to $f^{\prime}$ copies of $\pi$, is spinorial. From before, this implies that $f^{\prime}\cdot\frac{h}{2}$ is even, and $f^{\prime}g$ is congruent to [math] or $3\mod 4$. One can verify this $f^{\prime}g$ condition is equivalent to (2) being even. Thus (1),(2),(4), and (5) above are all even iff the restrictions of $\Pi$ to $S_{n}\times\{1\}$ and $\{1\}\times S_{n^{\prime}}$ are spinorial. Theorem 1.2 follows from this, since $ff^{\prime}=\deg\Pi$.

∎