Universität zu Köln

Montealegre-Mora, Felipe and Gross, David (2021). RANK-DEFICIENT REPRESENTATIONS IN THE THETA CORRESPONDENCE OVER FINITE FIELDS ARISE FROM QUANTUM CODES. Represent. Theory, 25. S. 193 - 224. PROVIDENCE: AMER MATHEMATICAL SOC. ISSN 1088-4165

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Abstract

Let V be a symplectic vector space and let mu be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation mu(circle times t) decomposes. If V is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible sub-representations of Sp(V) and the irreps of an orthogonal group O(t). It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp(V) representation. They show that a variant of the Theta correspondence continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp(V)-subrepresentations arise from embeddings of lower-order tensor products of mu and mu into mu(circle times t). The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp(V)-subrepresentations of mu(circle times t) are labelled by the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for r <= t. The results hold in odd charachteristic and the stable range t <= 1/2 dim V. Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Montealegre-Mora, Felipe UNSPECIFIED UNSPECIFIED UNSPECIFIED Gross, David UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-567373
DOI:	10.1090/ert/563
Journal or Publication Title:	Represent. Theory
Volume:	25
Page Range:	S. 193 - 224
Date:	2021
Publisher:	AMER MATHEMATICAL SOC
Place of Publication:	PROVIDENCE
ISSN:	1088-4165
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/56737