# Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions

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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