The Schur idempotent and the Schur functor for type-A q-Schur algebras

Let $R$ be a commutative ring with $1$, let $q\in R^{\times }$, let $d,n\ge 0$ with $d\ge n$, and let $H=H_{R,q}(S_n)$ (The generic type-A Hecke algebra). Write $S:=S(d,n)={\mathrm{E}\mathrm{n}\mathrm{d}}_H(M)$ for the type-A q-Schur algebra with $M={⨁}_{\mu \in \mathrm{\Lambda }(d,n)}{M}^{\mu }$ the direct sum of the permutation modules (The type-A q-Schur algebra and its Weyl modules, The direct sum of an indexed family of modules) and let $\{{\phi }_{ST}:\lambda \in {\mathrm{\Lambda }}^{+}(d,n),S,T\in T_0(\lambda )\}$ be its cellular semistandard basis (Murphy elements and Murphy tableau elements for the underlying Murphy elements and The standard basis of the generic type-A Hecke algebra for the standard basis along which this basis is defined). Put $\omega :=(1^n,0^{d-n})\in \mathrm{\Lambda }(d,n)$, the length-$d$ composition with $n$ entries equal to $1$ (legitimate because $d\ge n$), so that the Young subgroup $S_{\omega }$ is trivial and $m_{\omega }=1$; hence $M^{\omega }=m_{\omega }H=H$ is the regular right $H$-module. Let $t^{\omega }$ be the unique standard $\omega$-tableau, with entries $1,2,\dots ,n$ down its single column; it is in particular the unique row-standard $\omega$-tableau whose entries increase down the column. Set $e:={\phi }_{t^{\omega }{t}^{\omega }}\in S\mathrm{.}$

Then:

1. $e$ is the projection of $M$ onto the direct summand $M^{\omega }=H$ along the other summands $M^{\mu }$, $\mu \mathrm{\ne }\omega$; in particular $e^2=e$ and $eSe\cong {\mathrm{E}\mathrm{n}\mathrm{d}}_H(M^{\omega })={\mathrm{E}\mathrm{n}\mathrm{d}}_H(H),$ via the algebra isomorphism $\rho :H⟶eSe,h⟼{\rho }_h,{\rho }_h(x):=hx,$ left multiplication by $h$ on $M^{\omega }=H$ being a right $H$-module endomorphism. Here the isomorphism $eSe\cong {\mathrm{E}\mathrm{n}\mathrm{d}}_H(M^{\omega })$ sends $efe$ to $(efe){\mathrm{\mid }}_{M^{\omega }}$ and is inverted by extending an endomorphism of $M^{\omega }$ by zero on the other summands, and the identification ${\mathrm{E}\mathrm{n}\mathrm{d}}_H(H)\cong H$ sends an endomorphism to its value at $1$. By the construction of the semistandard basis elements ${\phi }_{ST}$ (Mathas (4.13) with Corollaries 4.9 and 4.12; see the locator above), the diagonal element ${\phi }_{t^{\omega }{t}^{\omega }}$ is exactly the endomorphism of $M$ that is the identity on $M^{\omega }$ and zero on all $M^{\mu }$ with $\mu \mathrm{\ne }\omega$, which is the projection asserted above.

2. The Schur functor is $F:\mathrm{m}\mathrm{o}\mathrm{d}\text{-}S⟶\mathrm{m}\mathrm{o}\mathrm{d}\text{-}H,F(X):=Xe=X{\otimes }_{S}Se,$ with the right $H$-action transported along $\rho$, so that $(xe)\cdot h:=x{\rho }_h$ for $x\in X$, $h\in H$ (Unital left and right modules over a ring; unqualified module means left module). It is exact and $R$-linear on morphisms: as a left $S$-module, $Se$ is the image of the left-$S$-linear projection $s\mapsto se$ on $S$, hence is a direct summand of the regular left module and is projective. Tensoring right $S$-modules with this left module is therefore exact. Equivalently, for a short exact sequence of right modules, an element $ye$ in the target lifts to $xe$ from any lift $x$ of $y$; kernels and injections are likewise preserved by the same idempotent projection. The bimodule corner action makes these maps right $H$-linear. The convention is fixed: the functor is $X\mapsto Xe$ on the right, and the left-module formula $X\mapsto eX$ is not used on this page.