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The Macdonald (q,t)(q,t)-permanent I: Algebra and positivity

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RA — Rings and Algebrasmath.RT — Representation Theorymath.FA — Functional Analysis

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We introduce the Macdonald (q,t)(q,t)-permanent, a two-parameter cycle-weighted permanent that contains the usual permanent and determinant, as well as the α\alpha-permanent of Vere-Jones, as degenerate cases. We prove a MacMahon-type master theorem for its generating function, and determine when it induces a valid kernel on tuples of vectors. Nonnegativity of the (q,t)(q,t)-permanent itself on semidefinite matrices turns out to be subtler: for t=qαt=q^\alpha nonnegativity holds exactly at the positive Wallach exponents; for a fixed qq we determine the region in rank two and for higher ranks under an additional condition, while for small qq positivity can even reach beyond the kernel region. Whether positivity forces t∈{0,q,q2,…}t\in\{0,q,q^2,\ldots\} remains an intriguing open problem.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The expansion of the author's notes into the present manuscript, together with its finite symbolic verification program, also used OpenAI's models, and Anthropic's Claude (Fable 5.1 and Opus 5.5) prepared this version of the manuscript. Assistant checks and reruns are not external peer review or formal verification. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT Pro
Anthropic
Claude Opus 5.5 and Fable 5.1

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Version history

  1. v1 · 28 pagesSubmitted by Suvrit SraInitial depositCurrentOct 31, 2026