Integral and field-valued Specht modules

Throughout, $R$ is a commutative ring, $n\ge 0$, and $\lambda ⊢n$ with Young diagram $[\lambda ]$. Tabloids, the left action $\sigma \cdot \{t\}:=\{\sigma \cdot t\}$ of $S_n$ on the finite set ${\mathrm{\Omega }}_{\lambda }$ of $\lambda$-tabloids, and the complex tabloid module $M^{\lambda }={\mathbb{C}}^{({\mathrm{\Omega }}_{\lambda })}$ are as in Young subgroups, tabloids, and permutation modules: rows are labelled and the order of entries inside a row is forgotten.

The tabloid module over $R$. Put $M_R^{\lambda }:=R^{({\mathrm{\Omega }}_{\lambda })},$ the free left $R$-module with the tabloids as $R$-basis, and let $S_n$ act on $M_R^{\lambda }$ by $R$-linear extension of $\sigma \cdot \{t\}:=\{\sigma \cdot t\}$ on basis elements. The row-set computation of Young subgroups, tabloids, and permutation modules shows that this rule is well defined on tabloids, and because the action on tableaux is a left action, $\mathrm{i}\mathrm{d}\cdot \{t\}=\{t\}$ and $\sigma \cdot (\tau \cdot \{t\})=(\sigma \tau )\cdot \{t\}$; hence $M_R^{\lambda }$ is a left module over $S_n$ and over the group ring $R[S_n]$.

Antisymmetrizers, polytabloids, and Specht modules. Let $t$ be a $\lambda$-tableau with column stabilizer $C_t$ (Row and column stabilizers), and let $\mathrm{sgn}:S_n\to \{+1,-1\}$ be the sign homomorphism (The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$); its values are read in $R$ through the unique unital ring homomorphism $\mathbb{Z}\to R$. Define ${\kappa }_t:={\sum }_{\gamma \in C_t}\mathrm{sgn}(\gamma )\gamma \in R[S_n],e_t:={\kappa }_t\cdot \{t\}={\sum }_{\gamma \in C_t}\mathrm{sgn}(\gamma )\{\gamma \cdot t\}\in M_R^{\lambda },$ and let $S_R^{\lambda }:={\mathrm{span}}_R\{e_t:t\text{ is a }\lambda \text{-tableau}\}\subseteq M_R^{\lambda }\mathrm{.}$ All sums here are finite sums over the finite group $C_t$, so ${\kappa }_t$, $e_t$ and $S_R^{\lambda }$ are well defined for every commutative ring $R$. For the empty partition the empty tableau is the unique one and has $C_t=\{1\}$, so ${\kappa }_t=1$, $e_t=\{\mathrm{\varnothing }\}$ and $S_R^{\mathrm{\varnothing }}=R$. For every tableau $t$ one has $C_t\cap R_t=\{1\}$: a permutation preserving every row set and every column set must fix the entry in each row-column intersection, since each intersection contains a single box. Hence the tabloids $\gamma \cdot \{t\}$, $\gamma \in C_t$, are distinct and the coefficient of $\{t\}$ in $e_t$ is $1$. Thus $e_t\mathrm{\ne }0$ whenever $R$ is nonzero; if $R$ is the zero ring, then $M_R^{\lambda }=S_R^{\lambda }=0$ and $e_t=0$.

Covariance of the construction over $R$. For every $\lambda$-tableau $t$ and every $\sigma \in S_n$, ${\kappa }_{\sigma \cdot t}=\sigma {\kappa }_t{\sigma }^{-1}\text{and}{e}_{\sigma \cdot t}=\sigma \cdot e_t,$ and for every $\gamma \in C_t$ one has $\gamma \cdot e_t=\mathrm{sgn}(\gamma )e_t$. Consequently $S_R^{\lambda }$ is an $R[S_n]$-submodule of $M_R^{\lambda }$, and for any single $\lambda$-tableau $t$ the orbit of $e_t$ spans $S_R^{\lambda }$. These identities are the ones published for $R=\mathbb{C}$ in Polytabloid covariance and the column sign rule; because the argument there consists only of reindexing the finite sums over $C_t$ and over $C_{\sigma \cdot t}$, it is valid verbatim over an arbitrary commutative ring. The two reindexings are written out in the remarks below.

Agreement with the complex Specht module. For $R=\mathbb{C}$ the module $M_{\mathbb{C}}^{\lambda }$, the element ${\kappa }_t$, the polytabloid $e_t$ and the space $S_{\mathbb{C}}^{\lambda }$ coincide with $M^{\lambda }$, ${\kappa }_t$, $e_t$ and $S^{\lambda }$ of Column antisymmetrizers, polytabloids, and Specht modules: the tabloid set and the left action are the same, the permutation sign used there is the sign of The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$ transported along the order-preserving relabelling described in that item, and the defining formulas are identical. In particular every published statement about ${\kappa }_t,e_t,S^{\lambda }$ applies to $S_{\mathbb{C}}^{\lambda }$.

• The two reindexings. First, $C_{\sigma \cdot t}=\sigma C_t{\sigma }^{-1}$: a permutation $\gamma$ preserves each column set $\sigma (B_j)$ of $\sigma \cdot t$ exactly when ${\sigma }^{-1}\gamma \sigma$ preserves each column set $B_j$ of $t$. Reindexing the finite sum defining ${\kappa }_{\sigma \cdot t}$ by $\gamma =\sigma c{\sigma }^{-1}$ and using $\mathrm{sgn}(\sigma c{\sigma }^{-1})=\mathrm{sgn}(c)$, which follows from multiplicativity of the sign and $\mathrm{sgn}({\sigma }^{-1})=\mathrm{sgn}(\sigma {)}^{-1}=\mathrm{sgn}(\sigma )$ in $\{+1,-1\}$, gives ${\kappa }_{\sigma \cdot t}=\sigma {\kappa }_t{\sigma }^{-1}$. Applying both sides to $\{\sigma \cdot t\}$ and using ${\sigma }^{-1}\cdot \{\sigma \cdot t\}=\{t\}$ gives $e_{\sigma \cdot t}=\sigma \cdot e_t$. Second, reindexing the sum defining ${\kappa }_t$ by $c\mapsto \gamma c$ for $\gamma \in C_t$ and using multiplicativity of the sign gives $\gamma {\kappa }_t=\mathrm{sgn}(\gamma ){\kappa }_t$, hence $\gamma \cdot e_t=\mathrm{sgn}(\gamma )e_t$.

• Reasons for stating the construction over a commutative ring. The restriction of a Specht module to $S_{n-1}$ has a removable-corner filtration over every field, and in positive characteristic that filtration need not split; the filtration and its modular failure are stated below on this page in terms of $S_F^{\lambda }$ for a general field $F$. The case $R=\mathbb{Z}$ records the integral lattice spanned by the polytabloids.

• No choice is used. The sums are over the finite group $C_t$, and the scalar extension $\mathbb{Z}\to R$ is unique, so no selection principle enters the definition.