Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite representation type. Is there a classification of the finite dimensional indecomposable $A$-modules (and the Auslander-Reiten quiver of $\text{mod}\,A$) in this case?
$\begingroup$
$\endgroup$
3
-
5$\begingroup$ I believe this is the same as representations of the Kronecker quiver arxiv.org/pdf/1209.4074.pdf $\endgroup$– Benjamin SteinbergCommented May 16, 2022 at 10:26
-
$\begingroup$ A quirk of $\mathbb{F}_2$ having characteristic $2$ is that $\mathbb{F}_2[C_2]$ is isomorphic to an exterior $\mathbb{F}_2$-algebra on one generator. Consequently $\mathbb{F}_2[C_2\times C_2]$ is isomorphic to an exterior $\mathbb{F}_2$-algebra on two generators. If you are willing to grade that algebra so that the exterior generators are homogeneous and in distinct positive degrees, then the fin. dim. indecomposable graded modules are classified by Adams in section III.16 of "Stable homotopy and generalized homology," for the sake of calculating topological $K$-theory using the Adams SS. $\endgroup$– user164898Commented May 16, 2022 at 16:14
-
2$\begingroup$ An older reference than Benjamin's is S. B. Conlon, J. Austral. Math. Soc. 10 (1969), 363-366. This is probably the first place where the indecomposables were classified over arbitrary fields $k$ of characteristic $2$. There are 4 infinite families $A_n$, $B_n$, $C_\pi(n)$ and $C_n(\infty)$ and one other $D$. The definition of $C_\pi(n)$ involves studying the irreducible polynomials over $k$. $\endgroup$– John MurrayCommented May 19, 2022 at 15:56
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
A complete description of the indecomposable modules for $C_2\times C_2$, including the Auslander-Reiten quiver is available in David Benson's book Modular Representation Theory: New Trends and Methods, Springer 1984, pp.176-181. This is over $\bar{\mathbb{F}}_2$; the result over arbitrary fields is given in the following few pages.
