no. 2
Representations and binomial coefficients
Fiebig, Peter1
1 FAU Erlangen–Nürnberg Department Mathematik Cauerstr. 11 91058 Erlangen Germany
Annals of Representation Theory, Volume 2 (2025) no. 2, pp. 249-279.

To a root system $R$ and a choice of coefficients in a field $K$ we associate a category $\mathcal{X}$ of graded spaces with operators. For an arbitrary choice of coefficients we show that we obtain a semisimple category in which the simple objects are parametrized by their highest weight. Then we assume that the coefficients are given by quantum binomials associated to $(K,q)$, where $q$ is an invertible element in $K$. In the case that $R$ is simply laced and $(K,q)$ has positive (quantum) characteristic, we construct a Frobenius pullback functor and prove a version of Steinberg’s tensor product theorem for $\mathcal{X}$. Then we prove that one can view the objects in $\mathcal{X}$ as the semisimple representations of Lusztig’s quantum group associated to $(R,K,q)$ (for $q=1$ we obtain semisimple representations of the hyperalgebra associated to $(R,K)$). Hence we obtain new proofs of the Frobenius and Steinberg theorems both in the representation theory of reductive algebraic groups and of quantum groups.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/art.24
Classification: 20G05, 20G42
Keywords: Representations of Algebraic groups and of quantum group representations, Frobenius pullback, Steinberg’s tensor product theorem

Fiebig, Peter 1

1 FAU Erlangen–Nürnberg Department Mathematik Cauerstr. 11 91058 Erlangen Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Fiebig, Peter. Representations and binomial coefficients. Annals of Representation Theory, Volume 2 (2025) no. 2, pp. 249-279. doi : 10.5802/art.24. https://art.centre-mersenne.org/articles/10.5802/art.24/

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