Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On tensor products of regular characters of the general linear and unitary groups of degree two over the principal ideal local rings of finite length

Archita Gupta, M Hassain, Pooja Singla

TL;DR

This work resolves the tensor product problem for regular irreducible representations of GL_2 and GU_2 over principal ideal local rings of finite length, classifying regular constituents into cuspidal, split semisimple, and split non-semisimple types. Using Mackey theory, orbit methods, and Heisenberg-type constructions, it derives sharp multiplicity bounds that depend on the types of the factors: at most 2 when two factors have distinct types, multiplicity-free cuspidal interactions, and a bound of len( rak{o})+1 in the ss-ss case (with equality only when the constituent is ss); sns-sns cases can yield multiplicities that grow with the residue field size when ll =2. A parallel field case is shown to have uniform multiplicity bounds by contrast. The paper also provides multiple constructions of regular representations (orbit-based, parity-aware, and ss/sns/cuspidal variants), describes the double coset structure S_{A_1}ackslash G / S_{A_2}, and proves detailed MF results that underpin the multiplicity bounds. Together, these results illuminate how tensor products behave over finite local rings versus finite fields, and open questions regarding residue-dependence and Ennola-type phenomena in this rank-two setting.

Abstract

Let $R$ be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Let $G(R)$ denote either the general linear group or the general unitary group of degree two over $R$. We study the decomposition of tensor products of irreducible representations of $G(R)$. It is known that the irreducible representations of $G(R)$ are built from regular representations, which are classified into three types: cuspidal, split semisimple, and split non-semisimple. We prove that the tensor product of any two regular irreducible representations of distinct types has irreducible constituents with multiplicity at most two. Moreover, we show that the regular part of the tensor product of a cuspidal representation with any other regular representation is multiplicity free. When both factors are of split semisimple type, we show that the multiplicity of any regular irreducible constituent is at most $\mathrm{length}(R) + 1$, and that this bound is achieved only when the constituent is also split semisimple. In contrast, we demonstrate that the multiplicity in the tensor product of two split non-semisimple representations can grow with the cardinality of the residue field when the length of the ring is at least two. In the case when $R$ is a finite field, all such tensor product multiplicities are uniformly bounded above by two. This highlights a significant difference between the behaviour of tensor products in the field case and in the more general finite local ring setting.

On tensor products of regular characters of the general linear and unitary groups of degree two over the principal ideal local rings of finite length

TL;DR

This work resolves the tensor product problem for regular irreducible representations of GL_2 and GU_2 over principal ideal local rings of finite length, classifying regular constituents into cuspidal, split semisimple, and split non-semisimple types. Using Mackey theory, orbit methods, and Heisenberg-type constructions, it derives sharp multiplicity bounds that depend on the types of the factors: at most 2 when two factors have distinct types, multiplicity-free cuspidal interactions, and a bound of len( rak{o})+1 in the ss-ss case (with equality only when the constituent is ss); sns-sns cases can yield multiplicities that grow with the residue field size when ll =2. A parallel field case is shown to have uniform multiplicity bounds by contrast. The paper also provides multiple constructions of regular representations (orbit-based, parity-aware, and ss/sns/cuspidal variants), describes the double coset structure S_{A_1}ackslash G / S_{A_2}, and proves detailed MF results that underpin the multiplicity bounds. Together, these results illuminate how tensor products behave over finite local rings versus finite fields, and open questions regarding residue-dependence and Ennola-type phenomena in this rank-two setting.

Abstract

Let be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Let denote either the general linear group or the general unitary group of degree two over . We study the decomposition of tensor products of irreducible representations of . It is known that the irreducible representations of are built from regular representations, which are classified into three types: cuspidal, split semisimple, and split non-semisimple. We prove that the tensor product of any two regular irreducible representations of distinct types has irreducible constituents with multiplicity at most two. Moreover, we show that the regular part of the tensor product of a cuspidal representation with any other regular representation is multiplicity free. When both factors are of split semisimple type, we show that the multiplicity of any regular irreducible constituent is at most , and that this bound is achieved only when the constituent is also split semisimple. In contrast, we demonstrate that the multiplicity in the tensor product of two split non-semisimple representations can grow with the cardinality of the residue field when the length of the ring is at least two. In the case when is a finite field, all such tensor product multiplicities are uniformly bounded above by two. This highlights a significant difference between the behaviour of tensor products in the field case and in the more general finite local ring setting.

Paper Structure

This paper contains 18 sections, 42 theorems, 140 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. The unitary group and its Lie algebra
  4. Proof of \ref{['thm:main-theorem']} and \ref{['thm:sns-theorem']} for $\ell = 1$
  5. Construction of regular representations of $\mathrm{G}(\mathfrak{o}_\ell)$
  6. Construction of regular representations of $\mathrm{G}(\mathfrak{o}_\ell)$ for $\ell$ even
  7. Construction of regular representations of $\mathrm{G}(\mathfrak{o}_\ell)$ for $\ell$ odd
  8. Alternate construction for split semisimple representations of $\mathrm{G}({\mathfrak{o}}_\ell)$
  9. Alternate construction for split non-semisimple representations of $\mathrm{G}({\mathfrak{o}}_\ell)$, $\ell$ odd.
  10. Alternate construction for cuspidal representations of $\mathrm{G}({\mathfrak{o}}_\ell)$, $\ell$ odd.
  11. Results related to the construction of representations of $\mathrm{G}(\mathfrak{o}_\ell)$
  12. Description of $S_{A_1} \backslash G /S_{A_2}$ for $\Xi_1, \Xi_2$ and $\Xi_3$
  13. Proof of \ref{['thm:main-theorem-2']}(1)-(3)
  14. Proof of \ref{['thm:main-theorem-2']}(1)-(2)
  15. Proof of \ref{['thm:main-theorem-2']}(3)
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\ell \geq 1$, and let $\lambda, \mu, \nu$ be regular irreducible representations of $\mathrm{G}(\mathfrak{o}_\ell)$.

Theorems & Definitions (76)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Lemma 4.1
  • ...and 66 more