Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes

TL;DR

The work addresses how tensor powers of the oscillator representation $\mu_V$ decompose over finite fields, introducing a rank notion that separates high-rank (Howe–Kashiwara–Vergne-like) from rank-deficient components. It shows that rank-deficient $\mathrm{Sp}(V)$-subrepresentations are precisely realized on tensor-power CSS codes, and provides an explicit decomposition of $\mu_{U\otimes V}$ into rank-labeled pieces tied to irreps of orthogonal groups via the η correspondence. A key contribution is proving that all rank-deficient components arise from embeddings of lower-order tensor powers into $\mu_V^{\otimes t}$ and that these embeddings are controlled by CSS-code structure, coset states, and Fourier-transform techniques. The results connect to Clifford-group invariants and offer a framework for understanding invariants and rank partitions in the finite-field setting, with future work extending to characteristic 2 and deeper Clifford-group connections.

Abstract

Let V be a symplectic vector space and let $μ$ be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation $μ^{\otimes t}$ decomposes. If V is a real vector space, then Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one correspondence between the irreducible subrepresentations 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 HKV duality 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 $μ$ and $\barμ$ into $μ^{\otimes t}$. 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 $μ^{\otimes t}$ 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.

Figures (3)

• Figure 1: Sketch of the commuting actions of the Weil representation and tensor-power CSS codes. The each tensor factor in the representation $\mu_V^{\otimes t}(S)$ for an arbitrary $S$ acts on a row (highlighted in blue). The code projector is an $n$-th tensor power of a projector supported on a column (red).
• Figure 2: Illustration of the various subspaces we will associate with an $F\in\mathop{\mathrm{Hom}}\nolimits(X\to U)$. In Lemma \ref{['lem:genesis']} and in the proof of the Main Theorem, the domain $X$ will be decomposed as a direct sum of $X_1=\ker F$ and some complement $X_2$. In Lemma \ref{['lem:dimension bound']} and in the proof of the Main Theorem, we decompose $U$ as a direct sum of $U_1=N_F=\mathop{\mathrm{range}}\nolimits F \cap (\mathop{\mathrm{range}}\nolimits F)^\perp$; $U_2$, some complement of $U_1$ within $\mathop{\mathrm{range}}\nolimits F$; and $U_3$, some complement of $\mathop{\mathrm{range}}\nolimits F$. These choices decompose $\mathop{\mathrm{Hom}}\nolimits(X\to U)$ into six different subspaces $\mathop{\mathrm{Hom}}\nolimits(X_i\to U_j)$, each of which can be visualized as a block in the matrix depicted. In Lemma \ref{['lem:genesis']}, the map $\Delta$ lives in the lower left hand side block, $\mathop{\mathrm{Hom}}\nolimits(X_1\to U_1=N_F)$. In Lemma \ref{['lem:trolling']}, we extend this to elements $\Delta=F-F'$ of the entire lower block $\mathop{\mathrm{Hom}}\nolimits(X\to U_1)$, subject to a rank constraint. In the proof of the Main Theorem, $G$ lives in the left block $\mathop{\mathrm{Hom}}\nolimits(X_1\to U)$. One could further subdivide $X_2$ into $X_2\cap F^{-1}(N_F)$ (left side of the dotted line), and some complement (right side of the dotted line). We do not make use of this division in our argument. With respect to this choice, $F$ is non-zero exactly on the two shaded blocks (where, in fact, it is invertible).
• Figure 3: "Branch and stem" structure of rank-$0$ subrepresentations of $\mu_{\mathbb{H}\otimes V}$ associated with the hyerpbolic plane. The vectors $f_1, f_2$ denote an orthogonal basis of $\mathbb{H}$. The red lines $I_\pm$ are the two isotropic spaces. A rank-deficient subrepresentation (Eq. \ref{['eqn:rank def intuition']}) of $\mu_{\mathbb{H},V}$ takes values that are constant on the "branches" $\{ F\;|\; N_F = I_\pm \}$, while the values add up on the "stem" $\{0\}$ where the spaces intersect.

Theorems & Definitions (33)

• Theorem 1.1: gh17
• Lemma : Lemma \ref{['lem:css representation']}, simplified version
• Theorem 1.2: Main Theorem
• Proposition : Proposition \ref{['prop:symplectoclifford']}, simplified version
• Lemma 2.1
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4