Hecke row permutation modules and ordinary Dipper-James polytabloids

Let $R$ be a commutative ring, let $q\in R^{\times }$, let $n\ge 0$, and let $H=H_{R,q}(S_n)$ be the type-A Hecke algebra over $R$ with generators $T_1,\dots ,T_{n-1}$, quadratic relations $(T_i-q)(T_i+1)=0$, braid and distant commutation relations, and standard basis $\{T_w:w\in S_n\}$ (The generic type-A Hecke algebra, The standard basis of the generic type-A Hecke algebra); here $\mathrm{\ell }(w)$ is the inversion length of $w\in S_n$ (Permutation Weyl group and inversion length). All modules below are right modules.

The star anti-involution. Let $\sigma \mapsto {\sigma }^{\ast }$ be the unique $R$-algebra anti-automorphism of $H$ with $T_i^{\ast }=T_i$ for every generator: it exists because reversal of words preserves the ideal of defining relations (the quadratic and braid relators are fixed, while a distant commutation relator is sent to its negative), so reversal descends from the free algebra to $H$, and it is unique because the generators generate $H$. On the standard basis $T_w^{\ast }=T_{w^{-1}}$, since reversing a reduced word for $w$ is a reduced word for $w^{-1}$.

Tableau conventions. Write $t\cdot \sigma$ for the right action of $S_n$ on tableaux defined by $(t\cdot \sigma )(c):={\sigma }^{-1}(t(c))$; it is the inverse of the left action $(\sigma \cdot t)(c)=\sigma (t(c))$ of Tableaux and standard tableaux, and it is a right action. A $\lambda$-tableau is row standard when its entries strictly increase along every row; columns are not constrained. Let $t^{\lambda }$ be the row-filled $\lambda$-tableau, whose row $i$ carries ${\lambda }_1+\cdots +{\lambda }_{i-1}+1,\dots ,{\lambda }_1+\cdots +{\lambda }_i$ from left to right, and let $t_{\lambda }$ be the column-filled $\lambda$-tableau, whose column $j$ carries consecutive integers downward; equivalently $t_{\lambda }(i,j)=t^{{\lambda }^{\mathrm{\prime }}}(j,i)$, so that $t_{\lambda }$ is the transpose of $t^{{\lambda }^{\mathrm{\prime }}}$ (Partitions, English diagrams, and conjugation).

The row subgroup and the coset representatives. Let $S_{\lambda }$ be the row subgroup of $\lambda$, the subgroup of $S_n$ preserving every row of $t^{\lambda }$; writing $B_i:=\{{\lambda }_1+\cdots +{\lambda }_{i-1}+1,{\lambda }_1+\cdots +{\lambda }_{i-1}+2,\dots ,{\lambda }_1+\cdots +{\lambda }_i\}$ for the $i$-th row block of $t^{\lambda }$, one has $S_{\lambda }=S(B_1)\times \cdots \times S(B_k)$, the direct product of the symmetric groups of the individual row blocks. Let $D_{\lambda }:=\{d\in S_n:t^{\lambda }\cdot d\text{ is row standard}\}\mathrm{.}$ This is the set of minimal-length right coset representatives of $S_{\lambda }$ in $S_n$: every right coset $S_{\lambda }d$ has exactly one element $d_0$ for which $t^{\lambda }\cdot d_0$ is row standard, and it satisfies $\mathrm{\ell }(w{d}_0)=\mathrm{\ell }(w)+\mathrm{\ell }(d_0)$ for all $w\in S_{\lambda }$ (The row permutation module: distinguished basis and Hecke action ↗). Let $w_{\lambda }\in S_n$ be the unique element with $t^{\lambda }\cdot w_{\lambda }=t_{\lambda }$; existence and uniqueness hold because the action of $S_n$ on $\lambda$-tableaux is free and transitive.

Generators and modules. For a partition $\lambda ⊢n$ put $x_{\lambda }:={\sum }_{w\in S_{\lambda }}{T}_w,y_{\lambda }:={\sum }_{w\in S_{\lambda }}(-q{)}^{-\mathrm{\ell }(w)}{T}_w,z_{\lambda }:=x_{\lambda }{T}_{w_{\lambda }}{y}_{{\lambda }^{\mathrm{\prime }}},$ where ${\lambda }^{\mathrm{\prime }}$ is the conjugate partition of $\lambda$ (Partitions, English diagrams, and conjugation). The row permutation module is the right ideal $M(\lambda ):=x_{\lambda }H\subseteq H$, and the ordinary Dipper-James polytabloid module is $P(\lambda ):=z_{\lambda }H\subseteq M(\lambda )$; $z_{\lambda }$ is its polytabloid generator.

The forms. The permutation form is the $R$-bilinear form $\beta$ on $M(\lambda )$ determined on the distinguished family $\{x_{\lambda }{T}_d:d\in D_{\lambda }\}$ by $\beta (x_{\lambda }{T}_d,x_{\lambda }{T}_e)={\delta }_{d,e}{q}^{\mathrm{\ell }(d)}(d,e\in D_{\lambda }),$ and the DJ form is the restriction of $\beta$ to $P(\lambda )$. That family is an $R$-basis of $M(\lambda )$ (The row permutation module: distinguished basis and Hecke action ↗), so the displayed values define a unique bilinear form; the family is $\{x_{\mathrm{\varnothing }}\}=\{1\}$ for the empty partition.

Small cases and base change. For $\lambda =\mathrm{\varnothing }$ set $x_{\mathrm{\varnothing }}=y_{\mathrm{\varnothing }}=z_{\mathrm{\varnothing }}=1$, so that $M(\mathrm{\varnothing })=P(\mathrm{\varnothing })=R$ and $\beta$ is the standard form; for $n=0,1$ there are no generators and $H=R$. Every base change $A=\mathbb{Z}[v^{\pm 1}]\to R$, $v\mapsto q$, with $q\in R^{\times }$, produces all objects above over $R$ by the same formulas; the suppliers of this page prove that the distinguished families and the form are compatible with such base change. No choice principle is used.

• Normalization dictionary. Fayers' Section 2.1 list of notation is the source of the displayed formulas, in the normalization $(T_i-q)(T_i+1)=0$ used here. Heard-Kujawa use the symmetric normalization $(T_i-q)(T_i+q^{-1})=0$ with $x_{\lambda }={\sum }_{w\in S_{\lambda }}{q}^{\mathrm{\ell }(w)}{T}_w$; their generator $z_{\lambda }=x_{\lambda }{T}_{{\sigma }_{\lambda }}{y}_{{\lambda }^{\mathsf{T}}}$ and their standard-basis indexing by standard tableaux of the transposed shape match the construction here, but the exact coefficients differ and only the combinatorial shape of the construction is taken from that source. Mathas' Chapter 3 uses the same normalization as this item.

• Orientation of the dictionaries. Fayers lets functions act on the right. Interpreting the source's label $j\sigma$ as ${\sigma }^{-1}(j)$ in the library's function-composition convention gives exactly the right tableau action used here. Reduced words in the adjacent transpositions then identify the algebra presentations and their coset and tableau indices.

• Definitional scope of the form. The permutation form is fixed here by its values on the distinguished basis and is a chosen contravariant form; its restriction to $P(\lambda )$ is taken as the definition of the DJ form. The historical Dipper-James self-form on $S^{\lambda }$ is this restriction, not an additional source claim, and nondegeneracy of $\beta$ or of its restriction is not asserted here.

• The empty and rank-zero cases. For $\lambda =\mathrm{\varnothing }$ the row group is $S_{\mathrm{\varnothing }}=S_0=\{1\}$, the group $D_{\mathrm{\varnothing }}=\{1\}$ is a singleton, and $M(\mathrm{\varnothing })=P(\mathrm{\varnothing })=R$ with basis $\{T_{\mathrm{i}\mathrm{d}}\}=\{1\}$, indexed by the identity of $S_0$.