Representations and cohomologies of the alternating group of degree 4

TL;DR

The paper addresses the problem of classifying integral representations of the alternating group $A_4$, with a focus on $2$-adic representations and the corresponding Tate cohomology of $A_4$-lattices. It develops a framework based on Bäckström orders and DR-valued graphs to describe the Auslander-Reiten quiver for $2$-adic representations, and uses globalization techniques to assemble local data into global indecomposable $A$-lattices. The main contributions are a complete local description of indecomposables at each prime, a glued AR-quiver for $2$-adic lattices, and explicit formulas for Tate cohomology $\\hat{H}^n(G,M)$ of all $A_4$-lattices, including both $2$-adic and $3$-adic cases, along with practical computational methods. These results have potential applications in crystallographic and Chernikov group classifications and demonstrate how adelic/global methods can streamline lattice-cohomology computations for finite groups. The work thereby provides concrete tools for lattice classification and cohomology calculations in the $A_4$ setting, with broader relevance to modular representation theory and group cohomology.

Abstract

We describe integral representations of the alternating group $A_4$, in particular, the Auslander-Reiten quiver of its 2-adic representations. Using these results we calculate Tate cohomologies of all $A_4$-lattices.

Theorems & Definitions (15)

• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• Theorem 2.8
• Example 2.9