Some Singular Vector-Valued Jack and Macdonald Polynomials

• Main Author: Charles F. Dunkl
• Format: Article
• Language: English
• Published: MDPI AG 2019-04-01
• Series: Symmetry
• Subjects: nonsymmetric Jack and Macdonald polynomials; singular values; Young tableaux; Hecke algebra
• Online Access: https://www.mdpi.com/2073-8994/11/4/503

For each partition <inline-formula> <math display="inline"> <semantics> <mi>&#964;</mi> </semantics> </math> </inline-formula> of <i>N</i>, there are irreducible modules of the symmetric groups <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">S</mi> <mi>N</mi> </msub> </semantics> </math> </inline-formula> and of the corresponding Hecke algebra <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">H</mi> <mi>N</mi> </msub> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> whose bases consist of the reverse standard Young tableaux of shape <inline-formula> <math display="inline"> <semantics> <mi>&#964;</mi> </semantics> </math> </inline-formula>. There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>N</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">S</mi> <mi>N</mi> </msub> </semantics> </math> </inline-formula> and the Hecke algebra <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">H</mi> <mi>N</mi> </msub> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mrow> </semantics> </math> </inline-formula>, there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by <inline-formula> <math display="inline"> <semantics> <mi>&#954;</mi> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>q</mi> <mo>,</mo> <mi>t</mi> </mfenced> </semantics> </math> </inline-formula>, respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>x</mi> <mrow> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mo>&otimes;</mo> <mi>S</mi> </mrow> </semantics> </math> </inline-formula>, where <i>S</i> is an arbitrary reverse standard Young tableau of shape <inline-formula> <math display="inline"> <semantics> <mi>&#964;</mi> </semantics> </math> </inline-formula>. The singular values depend on the properties of the edge of the Ferrers diagram of <inline-formula> <math display="inline"> <semantics> <mi>&#964;</mi> </semantics> </math> </inline-formula>.