Graded Characters of Modules Supported in the Closure of a Nilpotent Conjugacy Class

This is a combinatorial study of the Poincare polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka?Foulkes polynomials and are q -analogues of Littlewood?Richardson coefficients. The coefficients of two-column Macdonald?Kostka polynomials also occur as a special case. It is conjectured that these q -analogues are the generating function of so-called catabolizable tableaux with the charge statistic of Lascoux and Schutzenberger. A general approach for a proof is given, and is completed in certain special cases including the Kostka?Foulkes case. Catabolizable tableaux are used to prove a characterization of Lascoux and Schutzenberger for the image of the tableaux of a given content under the standardization map that preserves the cyclage poset.

[1]  Sergey Fomin,et al.  A Littlewood-Richardson Miscellany , 1993, Eur. J. Comb..

[2]  A. Kirillov,et al.  A Generalization of the Kostka–Foulkes Polynomials , 1998, math/9803062.

[3]  Alain,et al.  CRYSTAL GRAPHS AND q-ANALOGUES OF WEIGHT MULTIPLICITIES FOR THE ROOT SYSTEM A n ∗ , 1995 .

[4]  B. Broer.  Normality of Some Nilpotent Varieties and Cohomology of Line Bundles on the Cotangent Bundle of the Flag Variety , 1994 .

[5]  I. G. MacDonald,et al.  Notes on Schubert polynomials , 1991 .

[6]  Bruce E. Sagan,et al.  The symmetric group - representations, combinatorial algorithms, and symmetric functions , 2001, Wadsworth & Brooks / Cole mathematics series.

[7]  Peter Littelmann,et al.  A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras , 1994 .

[8]  Alain Lascoux,et al.  Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties , 1995 .

[9]  W. Hesselink.  Polarizations in the classical groups , 1978 .

[10]  J. R. Kline.  Memoirs of the American Mathematical Society , 1949 .

[11]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[12]  Dennis E. White.  Some Connections between the Littlewood-Richardson , 1981, J. Comb. Theory, Ser. A.

[13]  A. Zelevinsky,et al.  A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence , 1981 .

[14]  Mark D. Haiman.  On mixed insertion, symmetry, and shifted young tableaux , 1989, J. Comb. Theory, Ser. A.

[15]  A. Lascoux,et al.  Crystal graphs andq-analogues of weight multiplicities for the root systemAn , 1995 .

[16]  Jeffrey B. Remmel,et al.  A simple proof of the Littlewood-Richardson rule and applications , 1998, Discret. Math..

[17]  S. Fishel.  Statistics for special q, t-kostka polynomials , 1995 .

[18]  Anne Schilling,et al.  Inhomogeneous Lattice Paths, Generalized Kostka Polynomials and An−1 Supernomials , 1998, math/9802111.

[19]  A. Morris.  The characters of the groupGL(n, q) , 1963 .

[20]  J. Weyman.  The equations of conjugacy classes of nilpotent matrices , 1989 .

[21]  V. Reiner,et al.  %-avoiding, Northwest Shapes, and Peelable Tableaux , 2022 .

[22]  Michael Atiyah,et al.  Lie theory and geometry : in honor of Bertram Kostant , 1997, The Mathematical Gazette.

[23]  Peter Littelmann,et al.  Crystal Graphs and Young Tableaux , 1995 .

[24]  J. Weyman,et al.  Bases for Coordinate Rings of Conjugacy Classes of Nilpotent Matrices , 1999 .

[25]  J. Stembridge.  Some particular entries of the two-parameter Kostka matrix , 1994 .

[26]  Peter Littelmann,et al.  Paths and root operators in representation theory , 1995 .