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Let $KG$ be a group algebra of a finite group $G$ such that the characteristic of $K$ divides the group order.

Question: When is a block of a group algebra (or even the whole group aglebra) a Koszul algebra? Is there a classification?

For example in characteristic two, the group algebra $KG$ is isomorphic to $K[x]/(x^2)$ when $G$ has two elements and here the group algebra is a Koszul algebra with Koszul dual the polynomial ring $K[x]$.

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For a finite group $G$ and a field $K$ of characteristic $p$ dividing $|G|$, the group algebra $KG$ is Koszul if and only if $p=2$ and $G/O(G)$ is an elementary abelian $2$-group. This is because if $KG$ is Koszul then Ext over the cohomology ring is finite total dimension, so $H^*(G,K)$ is regular, and hence a polynomial ring. This happens if and only if $G/O(G)$ is an elementary abelian $2$-group, by a theorem I proved with Jon Carlson (MR1142778). Here, $O(G)$ denotes the largest odd order normal subgroup of $G$.

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  • $\begingroup$ Incidentally, when the group is not a $p$-group, I'm not quite sure how you want to interpret the word "Koszul", so maybe the answer is really if and only if $p=2$ and $G$ is an elementary abelian $2$-group. $\endgroup$
    – Dave Benson
    Commented Mar 26, 2023 at 13:42
  • $\begingroup$ What do you mean? I use the definition of Koszul algebra as in the article "KOSZUL DUALITY PATTERNS IN REPRESENTATION THEORY" by Beilinson, Ginzburg and Soergel, so that $A_0$ has just to be semisimple. $\endgroup$
    – Mare
    Commented Mar 26, 2023 at 16:04
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    $\begingroup$ They are using ${\mathbb Z}$-graded algebras with zero part semisimple. Are you applying their definition to the associated graded of the radical filtration? $\endgroup$
    – Dave Benson
    Commented Mar 26, 2023 at 16:47

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