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arXiv · 1710.02606

The Hilbert series of $\operatorname{SL}_2$-invariants

Pedro de Carvalho Cayres Pinto·Hans-Christian Herbig·Daniel Herden·Christopher Seaton
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Abstract

Let $V$ be a finite dimensional representations of the group $\operatorname{SL}_2$ of $2\times 2$ matrices with complex coefficients and determinant one. Let $R=\mathbb{C}[V]^{\operatorname{SL}_2}$ be the algebra of $\operatorname{SL}_2$-invariant polynomials on $V$. We present a calculation of the Hilbert series $\operatorname{Hilb}_R(t)=\sum_{n\ge 0}\dim (R_n)\: t^n$ as well as formulas for the first four coefficients of the Laurent expansion of $\operatorname{Hilb}_R(t)$ at $t=1$.

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Pedro de Carvalho Cayres Pinto, Hans-Christian Herbig, Daniel Herden, Christopher Seaton. 2017-10-06. The Hilbert series of $\operatorname{SL}_2$-invariants. https://doi.org/10.1142/s0219199719500172

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