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<p>In this paper, using crystal theory, we establish the existence of a new family of irreducible components arising in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac–Moody algebras. This work is motivated by the Schur positivity conjecture, Kostant’s conjecture, and Wahl’s conjecture. Furthermore, we prove the Schur positivity conjecture in full generality for finite-dimensional simple Lie algebras under the assumption that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda&gt;&gt; \mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>&gt;</mo> <mo>&gt;</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>; that is, if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> are the dominant weights in the tensor product, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\lambda +w\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>+</mo> <mi>w</mi> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> remains dominant for all <i>w</i> in the Weyl group.</p>

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Existence of a New Family of Irreducible Components in the Tensor Product and its Applications

  • 期刊论文

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  • 发表日期：

    2025年9月17日

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  • Rekha Biswal,
  • Stéphane Gaussent

摘要

In this paper, using crystal theory, we establish the existence of a new family of irreducible components arising in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac–Moody algebras. This work is motivated by the Schur positivity conjecture, Kostant’s conjecture, and Wahl’s conjecture. Furthermore, we prove the Schur positivity conjecture in full generality for finite-dimensional simple Lie algebras under the assumption that \(\lambda>> \mu \) λ > > μ ; that is, if \(\lambda \) λ and \(\mu \) μ are the dominant weights in the tensor product, then \(\lambda +w\mu \) λ + w μ remains dominant for all w in the Weyl group.