Nonzero modular Specht quotient criterion

The map $\phi :S_F^{\lambda }\to (S_F^{\lambda }{)}^{\ast }$, $\phi (v)={\beta }_F(v,\cdot ){\mathrm{\mid }}_{S_F^{\lambda }}$, is $F$-linear, and its kernel is exactly $R_F^{\lambda }$: an element $v\in S_F^{\lambda }$ lies in the kernel if and only if ${\beta }_F(v,s)=0$ for all $s\in S_F^{\lambda }$, that is, if and only if $v\in (S_F^{\lambda }{)}^{\perp }$. If $(e_{s_1},\dots ,e_{s_m})$ is the standard basis of $S_F^{\lambda }$ and $(e_{s_1}^{\ast },\dots ,e_{s_m}^{\ast })$ is its dual basis of $(S_F^{\lambda }{)}^{\ast }$, then the matrix of $\phi$ in these bases is $(\phi (e_{s_i})(e_{s_j}){)}_{i,j}=({\beta }_F(e_{s_i},e_{s_j}){)}_{i,j}$, which is the entrywise image in $F$ of the integer matrix $G_{\lambda }$ by [F2]. Hence $\mathrm{rank}\phi ={\mathrm{rank}}_F(G_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p)$ by [F7], and rank-nullity of $\phi$ gives ${\dim }_F{D}_F^{\lambda }={\dim }_F{S}_F^{\lambda }-{\dim }_F{R}_F^{\lambda }={\mathrm{rank}}_F(G_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p)\mathrm{.}$

A prime $p$ divides the gcd $g_{\lambda }$ of the entries $G_{ij}$ of $G_{\lambda }$ if and only if $p$ divides every entry $G_{ij}$; by [F4] this gcd is the same as the gcd of all integral pairings $\beta (e_s,e_t)$ of polytabloids. In particular, over $F$, the condition $p\mid g_{\lambda }$ is equivalent to the matrix $G_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p$ being the zero matrix.

By [F4] and [F6], $p\mid g_{\lambda }$ if and only if $\lambda$ is not $p$-regular, that is, if and only if $z_j(\lambda )\ge p$ for some $j\ge 1$.

By step 1.1, $D_F^{\lambda }=0$ if and only if ${\mathrm{rank}}_F(G_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p)=0$, which happens exactly when $G_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p$ is the zero matrix; by step 1.2 this is equivalent to $p\mid g_{\lambda }$, and hence to $p$ dividing every entry of $G_{\lambda }$.

Combining steps 2.1 and 1.3 gives the equivalences of assertion 1: $D_F^{\lambda }=0$ exactly when $p$ divides every entry of $G_{\lambda }$, exactly when $p\mid g_{\lambda }$, exactly when $\lambda$ is not $p$-regular, and equivalently $D_F^{\lambda }\mathrm{\ne }0$ exactly when $\lambda$ is $p$-regular. If $\lambda$ is $p$-regular, then $D_F^{\lambda }\mathrm{\ne }0$ by this equivalence, and [F5] applies in its nonzero case: $D_F^{\lambda }$ is self-dual and absolutely irreducible, $R_F^{\lambda }$ is the unique maximal submodule of $S_F^{\lambda }$ and equals $\mathrm{rad}(S_F^{\lambda })$, and $D_F^{\lambda }$ is the simple head of $S_F^{\lambda }$. This is assertion 3.

For assertion 2, step 1.1 gives ${\dim }_F{D}_F^{\lambda }={\mathrm{rank}}_F(G_{\lambda }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p)$ for every field $F$ of characteristic $p$; by step 3.1 this dimension is positive exactly when $\lambda$ is $p$-regular. The rank of the integer matrix $G_{\lambda }$ over $F$ is the largest $r$ for which some $r\times r$ minor of $G_{\lambda }$ has nonzero image in $F$; a minor is an integer and its image in $F$ is nonzero exactly when $p$ does not divide it, so this integer $r$ depends only on $\lambda$ and $p$ and not on the particular field $F$ of characteristic $p$. This common value is the $p$-rank of the Gram matrix. For the $k$ of a splitting $p$-modular system, [F3] exhibits the same value as ${\dim }_k{D}^{\lambda }$, so the statement over a general field specializes to the modular Specht quotient of the definition.

Assertions 1 and 2 are steps 3.1 and 4.1, and the structural part of assertion 3 is the second half of step 3.1; the structure statement is invoked only in the nonzero case, to which [F5] applies, and no simplicity of $S_F^{\lambda }$ is claimed. For $n=0$ and $\lambda =\mathrm{\varnothing }$ there is exactly one tabloid and one polytabloid $e_t$ with $C_t=\{1\}$ and ${\beta }_F(e_t,e_t)=1$, so $G_{\mathrm{\varnothing }}=(1)$, $g_{\mathrm{\varnothing }}=1$, $R_F^{\mathrm{\varnothing }}=0$ and $D_F^{\mathrm{\varnothing }}\cong F$ has dimension $1={\mathrm{rank}}_F(G_{\mathrm{\varnothing }}<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p)$; the empty partition is $p$-regular for every prime $p$ by [F6], so it is covered by the nonzero case and the criterion holds. No step divides by $p$ or by a group order, and no positivity or averaging is used, so characteristic $2$ is included without special treatment.