Timeline for Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$

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Oct 6, 2017 at 9:27	comment	added	Kasper Andersen		The paper "A study of certain modular representations" (ac.els-cdn.com/0021869378901163/…) proves a lot of results concerning the structure of $\text{Sym}^{g}(k^2)$, among other things that $\text{Sym}^{g+p (p-1)}(k^2) \cong \text{Sym}^g(k^2) \oplus \text{projective module}$.
Sep 22, 2017 at 13:53	comment	added	tkr		@WilberdvanderKallen Excellent point. It's worth pointing out that my basis naturally arose from considering $\alpha = Y^p-X^{p-1}Y$ as an element of $\mathrm{Sym}^p$. Multiplication by $\alpha$ induces an inclusion $\mathrm{Sym}^g \rightarrow \mathrm{Sym}^{g+p}$ which commutes with the operator $\Delta = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} - 1$. I used this to compute the cokernel of $\Delta$ inductively, which is the first step to calculating the $H_0(H,\mathrm{Sym}^g(k^2))$. Perhaps writing it in your form arises the same way; I need more coffee on Fridays.
Sep 22, 2017 at 9:05	comment	added	Wilberd van der Kallen		The span of the vectors $(Y^p-X^{p-1}Y)^iY^{g-pi}$ is better described as the span of the $(X^{p-1})^iY^{g-pi+i}$.
Sep 21, 2017 at 19:16	history	edited	tkr	CC BY-SA 3.0	edited title
Sep 21, 2017 at 19:11	history	asked	tkr	CC BY-SA 3.0