## Data

• Title: On a symplectic quantum Howe duality
• Authors: Elijah Bodish and Daniel Tubbenhauer
• Status: Preprint. Last update: Tue, 7 Mar 2023 17:25:21 EST
• ArXiv link: https://arxiv.org/abs/2303.04264
• Code on GitHub: Click and Click

## Abstract

We prove a nonsemisimple quantum version of Howe's duality with the rank 2n symplectic and the rank 2 special linear group acting on the exterior algebra of type C. We also discuss the first steps towards the symplectic analog of harmonic analysis on quantum spheres, give character formulas for various fundamental modules, and construct canonical bases of the exterior algebra.

## A few extra words

A cornerstone of modern invariant theory are Schur-Weyl-Brauer dualities, which allows one to play two commuting actions on tensor space against one another.
Building on earlier work, these dualities were generalised far beyond tensor space by Howe in the legendary 1995 Schur lectures. Howe's goal, that was achieved magnificently, was to present a new approach to classical invariant theory.
As Howe points out, classical invariant theory, historically speaking, was mostly about group actions on symmetric algebras or, less classical, on exterior algebras. Howe reformulated these actions in the double centralizer (or double commutant) approach of Schur-Weyl-Brauer dualities, and the outcome is what we call Howe dualities.
An example of a Howe duality (here formulated for the Lie algebra instead of the Lie group) reads as follows. Consider the $$\mathbb{C}$$-vector space $$\Lambda_{\mathbb{C}}:=\Lambda_{\mathbb{C}}(\mathbb{C}^{m}\otimes\mathbb{C}^{n})$$, i.e. the exterior algebra of $$\mathbb{C}^{m}\otimes\mathbb{C}^{n}$$. Then:

• There are commuting actions of $$U_{\mathbb{C}}(\mathfrak{gl}_{m})$$ and $$U_{\mathbb{C}}(\mathfrak{gl}_{n})$$ on $$\Lambda_{\mathbb{C}}$$.
• These actions generate each others centralizer.
• There is an explicit $$U_{\mathbb{C}}(\mathfrak{gl}_{m})\text{-}U_{\mathbb{C}}(\mathfrak{gl}_{n})^{op}$$-module decomposition of $$\Lambda_{\mathbb{C}}$$ pairing a Weyl module and a dual Weyl module.
This is known as exterior type A Howe duality. As Howe explains, when working over the complex numbers, the various forms of Howe dualities are equivalent to the respective Schur-Weyl-Brauer dualities. In particular, Howe-type dualities have been of crucial importance for invariant theory every since, but are also pervasive in other fields. Most notably, representation theory, low dimensional topology, and categorification.
The starting point of this paper is to take the easiest of Howe's dualities outside of type A and to prove it in the nonsemisimple and quantum case. It turns out that the easiest is the following, which is the main theorem of this paper. Let $$\mathcal{A}=\mathbb{Z}[q,q^{-1}]$$, and let $$\Lambda_{\mathcal{A}}$$ be the exterior algebra of the symplectic quantum vector representation.
• There are commuting actions of $$U_{\mathcal{A}}(\mathfrak{sp}_{2n})$$ and $$U_{\mathcal{A}}(\mathfrak{sl}_{2})$$ on $$\Lambda_{\mathcal{A}}$$.
• These actions generate each others centralizer.
• There is an explicit statement which, after specialization to the semisimple case, gives a $$U_{\mathcal{A}}(\mathfrak{sp}_{2n})\text{-}U_{\mathcal{A}}(\mathfrak{sl}_{2})$$-module decomposition of $$\Lambda_{\mathcal{A}}$$ pairing a Weyl module and a dual Weyl module.
An upshot of this is an explicit description for fundamental tilting modules to be simple, and we even give their Weyl characters. For example, for $$n=78$$ and quantum characteristic $$p=7,\ell=3$$ and $$p=3,\ell=2$$, respectively, we get:
The columns are the fundamental tilting modules $$T(\varpi_{i-1})$$ and the rows the fundamental Weyl modules $$\Delta(\varpi_{j-1})$$, both starting to count at one.