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arXiv · 2610.37185

On Some Lie Group Invariant noncommutative functions

Jiachen Lu
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Abstract

We study noncommutative polynomials and functions invariant under linear actions of infinite (compact Lie) groups, with a focus on $SU(2)$ acting on two noncommuting variables. For $SU(2)$, we give a complete description of the invariant noncommutative polynomials: there are no invariants of odd degree, and in degree $2n$ the invariants are exactly the span of the $S_{2n}$-orbit of $(XY-YX)^n$. We also construct an explicit basis recursively from the operation $(A,B)\mapsto XAYB-YAXB$, and show that the space of degree-$2n$ invariants has dimension equal to the $n$-th Catalan number. All proofs proceed by direct computation with the Haar integral.

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Jiachen Lu. 2026-10-27. On Some Lie Group Invariant noncommutative functions. https://arxiv.org/abs/2610.37185

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