A charge monomial basis of the Garsia-Procesi ring

We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in [3]. Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mover accent="true"> <mml:mi>H</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mi>μ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> with respect to a Young subgroup <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> </mml:math> is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> is given by the catabolizability formula for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mover accent="true"> <mml:mi>H</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mi>μ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .