The Classification of All Singular Nonsymmetric Macdonald Polynomials
The affine Hecke algebra of type <i>A</i> has two parameters <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfenced close=")" open="("><mi>q</mi><mo>,</mo&g...
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MDPI AG
2022-04-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/5/208 |
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| author | Charles F. Dunkl |
| author_facet | Charles F. Dunkl |
| author_sort | Charles F. Dunkl |
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| description | The affine Hecke algebra of type <i>A</i> has two parameters <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfenced close=")" open="("><mi>q</mi><mo>,</mo><mi>t</mi></mfenced></semantics></math></inline-formula> and acts on polynomials in <i>N</i> variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>q</mi><mi>m</mi></msup><mo>=</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>N</mi><mo>.</mo></mrow></semantics></math></inline-formula> For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of <i>N</i>) of the Hecke algebra module is <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfenced close=")" open="("><mi>d</mi><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>r</mi></mfenced></semantics></math></inline-formula> for some <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In the present paper, it is shown that there are no other singular polynomials. |
| first_indexed | 2024-03-10T03:20:25Z |
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| spelling | doaj.art-64813066182245689480bf677e1de6cd2023-11-23T10:04:05ZengMDPI AGAxioms2075-16802022-04-0111520810.3390/axioms11050208The Classification of All Singular Nonsymmetric Macdonald PolynomialsCharles F. Dunkl0Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USAThe affine Hecke algebra of type <i>A</i> has two parameters <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfenced close=")" open="("><mi>q</mi><mo>,</mo><mi>t</mi></mfenced></semantics></math></inline-formula> and acts on polynomials in <i>N</i> variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>q</mi><mi>m</mi></msup><mo>=</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>N</mi><mo>.</mo></mrow></semantics></math></inline-formula> For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of <i>N</i>) of the Hecke algebra module is <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfenced close=")" open="("><mi>d</mi><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>r</mi></mfenced></semantics></math></inline-formula> for some <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In the present paper, it is shown that there are no other singular polynomials.https://www.mdpi.com/2075-1680/11/5/208nonsymmetric Macdonald polynomialsthe affine Hecke algebra of type <i>A</i>Young tableauxJucys–Murphy operators |
| spellingShingle | Charles F. Dunkl The Classification of All Singular Nonsymmetric Macdonald Polynomials Axioms nonsymmetric Macdonald polynomials the affine Hecke algebra of type <i>A</i> Young tableaux Jucys–Murphy operators |
| title | The Classification of All Singular Nonsymmetric Macdonald Polynomials |
| title_full | The Classification of All Singular Nonsymmetric Macdonald Polynomials |
| title_fullStr | The Classification of All Singular Nonsymmetric Macdonald Polynomials |
| title_full_unstemmed | The Classification of All Singular Nonsymmetric Macdonald Polynomials |
| title_short | The Classification of All Singular Nonsymmetric Macdonald Polynomials |
| title_sort | classification of all singular nonsymmetric macdonald polynomials |
| topic | nonsymmetric Macdonald polynomials the affine Hecke algebra of type <i>A</i> Young tableaux Jucys–Murphy operators |
| url | https://www.mdpi.com/2075-1680/11/5/208 |
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