On representations of general linear groups over principal ideal local rings of length two

Abstract

We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of all general linear groups over principal ideal local rings of arbitrary length in the function field case.

On Representations of Classical Groups Over Principal Ideal Local Rings of Length Two

“…The bilinear form ., . : [7] to remove any ambiguity) denote the stabilizer of ψ A in GL n (O 2 ). In this section we prove Proposition 2.2 for Orthogonal, Unitary, Symplectic and Special linear groups(p ∤ n).…”

“…This proves Step 2. For step 3, first of all we briefly recall the construction of character χ G ψ A of T G (ψ A ) that extends ψ A , from Singla [7]. For a matrix A as given in (4.1), let Z GL n (O 2 ) (s(A)) = {g ∈ GL n (O 2 ) | gs(A) = s(A)g} and m ij be the size of each matrix A ij .…”

Finite-field consensus

“…, 0}. In other words, the eigenvalues of a finite-field matrix achieving consensus are all contained in the considered finite field and, consequently, every finite-field matrix achieving consensus can be represented in Jordan canonical form; see [26] and [27,Theorem 3.5]. We conclude this section by characterizing the convergence value of a finite-field consensus network.…”

On invariant factors of primary components of square matrices

“…For proof of these see [[3], A.VII.31], [6]. Theorem 3.9 (Primary Decomposition) Every matrix A ∈ M n (K) is similar to a matrix of the form ⊕ P A P .…”