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<p>The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group <i>W</i> implies that there is a similar “extremal weight property” for every weight obtained by applying an element of <i>W</i> to the highest weight. In this paper, we conjecture an analogous “extremal monomial property” of the <i>q</i>-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on <i>q</i>-characters defined by Chari. In the case of the identity element of <i>W</i>, this is the highest monomial property of <i>q</i>-characters proved in [<CitationRef CitationID="CR13">13</CitationRef>]. Here, we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(w \in W\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>∈</mo> <mi>W</mi> </mrow> </math></EquationSource> </InlineEquation> turns out to be equivalent to polynomiality of the “<i>X</i>-series” corresponding to <i>w</i>, which we introduce in this paper. We show that these <i>X</i>-series are equal to certain limits of the generalized Baxter operators for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(w \in W\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>∈</mo> <mi>W</mi> </mrow> </math></EquationSource> </InlineEquation>. Thus, we find a new bridge between <i>q</i>-characters and the spectra of XXZ-type quantum integrable models associated with quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of [<CitationRef CitationID="CR8">8</CitationRef>].</p>

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Extremal monomial property of q-characters and polynomiality of the X-series

  • 期刊论文

  • 发表日期：

    2025年9月20日

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  • Edward Frenkel,
  • David Hernandez

摘要

The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group W implies that there is a similar “extremal weight property” for every weight obtained by applying an element of W to the highest weight. In this paper, we conjecture an analogous “extremal monomial property” of the q-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on q-characters defined by Chari. In the case of the identity element of W, this is the highest monomial property of q-characters proved in [13]. Here, we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each \(w \in W\) w ∈ W turns out to be equivalent to polynomiality of the “X-series” corresponding to w, which we introduce in this paper. We show that these X-series are equal to certain limits of the generalized Baxter operators for all \(w \in W\) w ∈ W . Thus, we find a new bridge between q-characters and the spectra of XXZ-type quantum integrable models associated with quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of [8].