Catalan functions and 𝑘-Schur positivity

Blasiak, Jonah¹ ; Morse, Jennifer² ; Pun, Anna¹ ; Summers, Daniel¹

¹ Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
² Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Journal of the American Mathematical Society, Tome 32 (2019) no. 4, pp. 921-963

Résumé

We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.

Détail

DOI : 10.1090/jams/921

Bibliographie

[1] Assaf, Sami H., Billey, Sara C. Affine dual equivalence and 𝑘-Schur functions J. Comb. 2012 343 399

[2] Blasiak, Jonah, Fomin, Sergey Noncommutative Schur functions, switchboards, and Schur positivity Selecta Math. (N.S.) 2017 727 766

[3] Broer, Abraham A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles J. Reine Angew. Math. 1997 153 169

[4] Broer, Bram Line bundles on the cotangent bundle of the flag variety Invent. Math. 1993 1 20

[5] Broer, Bram Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety 1994 1 19

[6] Carlsson, Erik, Mellit, Anton A proof of the shuffle conjecture J. Amer. Math. Soc. 2018 661 697

[7] Chen, Li-Chung Skew-Linked Partitions and a Representation-Theoretic Model for k-Schur Functions 2010 71

[8] Désarménien, J., Leclerc, B., Thibon, J.-Y. Hall-Littlewood functions and Kostka-Foulkes polynomials in representation theory Sém. Lothar. Combin. 1994

[9] Fomin, Sergey, Greene, Curtis Noncommutative Schur functions and their applications Discrete Math. 1998 179 200

[10] Garsia, Adriano M. Orthogonality of Milne’s polynomials and raising operators Discrete Math. 1992 247 264

[11] Garsia, A. M., Procesi, C. On certain graded 𝑆_{𝑛}-modules and the 𝑞-Kostka polynomials Adv. Math. 1992 82 138

[12] Haglund, J., Haiman, M., Loehr, N., Remmel, J. B., Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants Duke Math. J. 2005 195 232

[13] Haiman, Mark Hilbert schemes, polygraphs and the Macdonald positivity conjecture J. Amer. Math. Soc. 2001 941 1006

[14] Haiman, Mark Combinatorics, symmetric functions, and Hilbert schemes 2003 39 111

[15] Hesselink, Wim H. Cohomology and the resolution of the nilpotent variety Math. Ann. 1976 249 252

[16] Jing, Nai Huan Vertex operators and Hall-Littlewood symmetric functions Adv. Math. 1991 226 248

[17] Kamnitzer, Joel Geometric constructions of the irreducible representations of 𝐺𝐿_{𝑛} 2011 1 18

[18] Kirillov, Anatol N., Schilling, Anne, Shimozono, Mark A bijection between Littlewood-Richardson tableaux and rigged configurations Selecta Math. (N.S.) 2002 67 135

[19] Lam, Thomas Schubert polynomials for the affine Grassmannian J. Amer. Math. Soc. 2008 259 281

[20] Lam, Thomas Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras Bull. Lond. Math. Soc. 2011 328 334

[21] Lam, Thomas, Lapointe, Luc, Morse, Jennifer, Shimozono, Mark Affine insertion and Pieri rules for the affine Grassmannian Mem. Amer. Math. Soc. 2010

[22] Lam, Thomas, Lapointe, Luc, Morse, Jennifer, Shimozono, Mark The poset of 𝑘-shapes and branching rules for 𝑘-Schur functions Mem. Amer. Math. Soc. 2013

[23] Lapointe, L., Lascoux, A., Morse, J. Tableau atoms and a new Macdonald positivity conjecture Duke Math. J. 2003 103 146

[24] Lapointe, L., Morse, J. Schur function analogs for a filtration of the symmetric function space J. Combin. Theory Ser. A 2003 191 224

[25] Lapointe, Luc, Morse, Jennifer Tableaux on 𝑘+1-cores, reduced words for affine permutations, and 𝑘-Schur expansions J. Combin. Theory Ser. A 2005 44 81

[26] Lapointe, Luc, Morse, Jennifer A 𝑘-tableau characterization of 𝑘-Schur functions Adv. Math. 2007 183 204

[27] Lapointe, Luc, Morse, Jennifer Quantum cohomology and the 𝑘-Schur basis Trans. Amer. Math. Soc. 2008 2021 2040

[28] Lusztig, George Some examples of square integrable representations of semisimple 𝑝-adic groups Trans. Amer. Math. Soc. 1983 623 653

[29] Macdonald, I. G. Symmetric functions and Hall polynomials 1995

[30] Panyushev, Dmitri I. Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles Selecta Math. (N.S.) 2010 315 342

[31] Schilling, Anne, Warnaar, S. Ole Inhomogeneous lattice paths, generalized Kostka polynomials and 𝐴_{𝑛-1} supernomials Comm. Math. Phys. 1999 359 401

[32] Shimozono, Mark A cyclage poset structure for Littlewood-Richardson tableaux European J. Combin. 2001 365 393

[33] Shimozono, Mark Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties J. Algebraic Combin. 2002 151 187

[34] Shimozono, Mark, Weyman, Jerzy Graded characters of modules supported in the closure of a nilpotent conjugacy class European J. Combin. 2000 257 288

[35] Shimozono, Mark, Zabrocki, Mike Hall-Littlewood vertex operators and generalized Kostka polynomials Adv. Math. 2001 66 85

[36] Stanley, Richard P. Enumerative combinatorics. Vol. 2 1999

[37] Zabrocki, Michael Alan On the action of the Hall-Littlewood vertex operator 1998 58