# zbMATH — the first resource for mathematics

Catalan functions and $$k$$-Schur positivity. (English) Zbl 1423.05192
Summary: 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.

##### MSC:
 500000 Symmetric functions and generalizations 5e+10 Combinatorial aspects of representation theory
Full Text:
##### References:
 [1] Assaf, Sami H.; Billey, Sara C., Affine dual equivalence and $$k$$-Schur functions, J. Comb., 3, 3, 343-399 (2012) · Zbl 1291.05204 [2] Blasiak, Jonah; Fomin, Sergey, Noncommutative Schur functions, switchboards, and Schur positivity, Selecta Math. (N.S.), 23, 1, 727-766 (2017) · Zbl 1355.05249 [3] Broer, Abraham, A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles, J. Reine Angew. Math., 493, 153-169 (1997) · Zbl 0911.14003 [4] Broer, Bram, Line bundles on the cotangent bundle of the flag variety, Invent. Math., 113, 1, 1-20 (1993) · Zbl 0807.14043 [5] Broer, Bram, Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety. Lie theory and geometry, Progr. Math. 123, 1-19 (1994), Birkh\"{a}user Boston, Boston, MA · Zbl 0855.22015 [6] Carlsson, Erik; Mellit, Anton, A proof of the shuffle conjecture, J. Amer. Math. Soc., 31, 3, 661-697 (2018) · Zbl 1387.05265 [7] Chen, Li-Chung, Skew-Linked Partitions and a Representation-Theoretic Model for k-Schur Functions, 71 pp. (2010), ProQuest LLC, Ann Arbor, MI [8] D\'{e}sarm\'{e}nien, J.; Leclerc, B.; Thibon, J.-Y., Hall-Littlewood functions and Kostka-Foulkes polynomials in representation theory, S\'{e}m. Lothar. Combin., 32, Art. B32c, approx. 38 pp. (1994) · Zbl 0855.05100 [9] Fomin, Sergey; Greene, Curtis, Noncommutative Schur functions and their applications, Discrete Math., 193, 1-3, 179-200 (1998) · Zbl 1011.05062 [10] Garsia, Adriano M., Orthogonality of Milne’s polynomials and raising operators, Discrete Math., 99, 1-3, 247-264 (1992) · Zbl 0763.05100 [11] Garsia, A. M.; Procesi, C., On certain graded $$S_n$$-modules and the $$q$$-Kostka polynomials, Adv. Math., 94, 1, 82-138 (1992) · Zbl 0797.20012 [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., 126, 2, 195-232 (2005) · Zbl 1069.05077 [13] Haiman, Mark, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc., 14, 4, 941-1006 (2001) · Zbl 1009.14001 [14] Haiman, Mark, Combinatorics, symmetric functions, and Hilbert schemes. Current developments in mathematics, 2002, 39-111 (2003), Int. Press, Somerville, MA · Zbl 1053.05118 [15] Hesselink, Wim H., Cohomology and the resolution of the nilpotent variety, Math. Ann., 223, 3, 249-252 (1976) · Zbl 0318.14007 [16] Jing, Nai Huan, Vertex operators and Hall-Littlewood symmetric functions, Adv. Math., 87, 2, 226-248 (1991) · Zbl 0742.16014 [17] Kamnitzer, Joel, Geometric constructions of the irreducible representations of $$GL_n$$. Geometric representation theory and extended affine Lie algebras, Fields Inst. Commun. 59, 1-18 (2011), Amer. Math. Soc., Providence, RI · Zbl 1229.22002 [18] Kirillov, Anatol N.; Schilling, Anne; Shimozono, Mark, A bijection between Littlewood-Richardson tableaux and rigged configurations, Selecta Math. (N.S.), 8, 1, 67-135 (2002) · Zbl 0986.05013 [19] Lam, Thomas, Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc., 21, 1, 259-281 (2008) · Zbl 1149.05045 [20] Lam, Thomas, Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras, Bull. Lond. Math. Soc., 43, 2, 328-334 (2011) · Zbl 1235.05152 [21] Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Shimozono, Mark, Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc., 208, 977, xii+82 pp. (2010) · Zbl 1208.14002 [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., 223, 1050, vi+101 pp. (2013) · Zbl 1292.05258 [23] Lapointe, L.; Lascoux, A.; Morse, J., Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J., 116, 1, 103-146 (2003) · Zbl 1020.05069 [24] Lapointe, L.; Morse, J., Schur function analogs for a filtration of the symmetric function space, J. Combin. Theory Ser. A, 101, 2, 191-224 (2003) · Zbl 1018.05101 [25] Lapointe, Luc; Morse, Jennifer, Tableaux on $$k+1$$-cores, reduced words for affine permutations, and $$k$$-Schur expansions, J. Combin. Theory Ser. A, 112, 1, 44-81 (2005) · Zbl 1120.05093 [26] Lapointe, Luc; Morse, Jennifer, A $$k$$-tableau characterization of $$k$$-Schur functions, Adv. Math., 213, 1, 183-204 (2007) · Zbl 1118.05096 [27] Lapointe, Luc; Morse, Jennifer, Quantum cohomology and the $$k$$-Schur basis, Trans. Amer. Math. Soc., 360, 4, 2021-2040 (2008) · Zbl 1132.05060 [28] Lusztig, George, Some examples of square integrable representations of semisimple $$p$$-adic groups, Trans. Amer. Math. Soc., 277, 2, 623-653 (1983) · Zbl 0526.22015 [29] Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, x+475 pp. (1995), The Clarendon Press, Oxford University Press, New York · Zbl 0899.05068 [30] Panyushev, Dmitri I., Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles, Selecta Math. (N.S.), 16, 2, 315-342 (2010) · Zbl 1248.17006 [31] Schilling, Anne; Warnaar, S. Ole, Inhomogeneous lattice paths, generalized Kostka polynomials and $$A_{n-1}$$ supernomials, Comm. Math. Phys., 202, 2, 359-401 (1999) · Zbl 0935.05090 [32] Shimozono, Mark, A cyclage poset structure for Littlewood-Richardson tableaux, European J. Combin., 22, 3, 365-393 (2001) · Zbl 0979.05107 [33] Shimozono, Mark, Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin., 15, 2, 151-187 (2002) · Zbl 1106.17305 [34] Shimozono, Mark; Weyman, Jerzy, Graded characters of modules supported in the closure of a nilpotent conjugacy class, European J. Combin., 21, 2, 257-288 (2000) · Zbl 0956.05100 [35] Shimozono, Mark; Zabrocki, Mike, Hall-Littlewood vertex operators and generalized Kostka polynomials, Adv. Math., 158, 1, 66-85 (2001) · Zbl 0997.17017 [36] Stanley, Richard P., Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics 62, xii+581 pp. (1999), Cambridge University Press, Cambridge · Zbl 0928.05001 [37] Zabrocki, Michael Alan, On the action of the Hall-Littlewood vertex operator, 58 pp. (1998), ProQuest LLC, Ann Arbor, MI
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.