Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted $G(\zeta)$ and $K(\zeta)$, where $$ G(\zeta)=\bigoplus_{n\geq0}G_n(\zeta) \text{ and } K(\zeta)=\bigoplus_{n\geq0}K_n(\zeta). $$ Here $G_n(\zeta)$ is the Grothendieck group of the category of finitely generated left $H_n(\zeta)$-modules, and $K_n(\zeta)$ is the Grothendieck group of finitely generated projective left $H_n(\zeta)$-modules. Induction and restriction endow these spaces with a bialgebra structure, which can be enhanced to a Hopf structure.

In "Hecke algebras at roots of unity and crystal cases of quantum affine algebras" Lascoux, Leclerc, and Thibon mention isomorphisms $$ G(\zeta) \cong Sym/I \text{ and } K(\zeta) \cong I^{\perp} $$ where $Sym$ is the algebra of symmetric functions, $I$ is the ideal generated by power sum symmetric functions $p_\ell,p_{2\ell},p_{3\ell},...$, and the complement is taken with respect to the standard inner product on $Sym$ (see page 215 of [LLT]).

My question: does anybody know a reference where this fact appears with proof? In [LLT] there is neither reference nor proof.


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The only place where I have seen this in the literature is by combining (15) on page 109 of Donkin's "The q-Schur algebra" with the usual results for the symmetric group which I assume can be found in Macdonald. Donkin doesn't consider the projective Grothendieck groups but once you have one isomorphism the Cartan pairing should give you the other.


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