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Hall-Littlewood-positive harmonic functionals on the algebra of symmetric functions

Cesar Cuenca, Grigori Olshanski

Abstract

We study the problem of describing the set of real functionals on the quotient $\textrm{Sym}/(p_2-1)$ of the ring of symmetric functions that are nonnegative on the images of certain modified Hall-Littlewood symmetric functions. This question is equivalent to the problem, posed in [Adv Math 395, p.108087 (2022)], of describing the set of coadjoint-invariant measures for unitary groups over a finite field in the infinite-dimensional setting. Our main results constitute partial progress towards this problem. Firstly, we show that the desired set of functionals is very large, in the sense that it contains explicit families of examples depending on infinitely many parameters. Secondly, we provide an analogue of Kerov's mixing construction that produces new sought after functionals from known old ones. This construction depends on an explicit "$p_2$-twisted action" of $\textrm{Sym}$ on itself and the resulting dual map that makes $\textrm{Sym}$ into a comodule. Finally, our third main result explains the relation between the $p_2$-twisted comultiplication and the usual comultiplication on $\textrm{Sym}$.

Hall-Littlewood-positive harmonic functionals on the algebra of symmetric functions

Abstract

We study the problem of describing the set of real functionals on the quotient of the ring of symmetric functions that are nonnegative on the images of certain modified Hall-Littlewood symmetric functions. This question is equivalent to the problem, posed in [Adv Math 395, p.108087 (2022)], of describing the set of coadjoint-invariant measures for unitary groups over a finite field in the infinite-dimensional setting. Our main results constitute partial progress towards this problem. Firstly, we show that the desired set of functionals is very large, in the sense that it contains explicit families of examples depending on infinitely many parameters. Secondly, we provide an analogue of Kerov's mixing construction that produces new sought after functionals from known old ones. This construction depends on an explicit "-twisted action" of on itself and the resulting dual map that makes into a comodule. Finally, our third main result explains the relation between the -twisted comultiplication and the usual comultiplication on .

Paper Structure

This paper contains 8 sections, 21 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries: branching graphs and Kerov's mixing construction
  4. The branching graphs $\widetilde{\mathbb Y}^{\mathrm{HL}}_{\mathrm{even}}(-t)$ and $\widetilde{\mathbb Y}^{\mathrm{HL}}_{\mathrm{odd}}(-t)$, and a theorem of Chen and Van Peski
  5. Embeddings $\Phi(t^2)\to\Psi_{\mathrm{even}}(-t)$ and $\Phi(t^2)\to\Psi_{\mathrm{odd}}(-t)$
  6. Analog of Kerov's mixing construction
  7. Interaction between $\Delta$ and $\widetilde{\Delta}$
  8. Concluding remarks

Key Result

Theorem 1.2

Fix $t\in(-1,1)$. The functionals $\varphi\in\operatorname{ex}(\Phi_1(t))$ are precisely those multiplicative functionals whose values on the generators $p_1,p_2,\dots$ are given by the formulas for some real parameters $\alpha_i$ and $\beta_j$ that satisfy the following conditions: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2: Kerov-Matveev
  • Definition 1.3: cf. Definition \ref{['def1.A']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 38 more