Wreath Macdonald polynomials at 𝑞=𝑡 as characters of rational Cherednik algebras

Mathiä, Dario; Thiel, Ulrich

doi:10.1090/tran/8774

Wreath Macdonald polynomials at $q=t$ as characters of rational Cherednik algebras
HTML articles powered by AMS MathViewer

by Dario Mathiä and Ulrich Thiel PDF

Trans. Amer. Math. Soc. 375 (2022), 8945-8968 Request permission

Abstract:

Using the theory of Macdonald [Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group $\mathfrak {S}_n$ are given by plethystically transformed Macdonald polynomials specialized at $q=t$. We generalize this to restricted rational Cherednik algebras of wreath product groups $C_\ell \wr \mathfrak {S}_n$ and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [Combinatorics, symmetric functions, and Hilbert schemes, Int. Press, Somerville, MA, 2003, pp. 39–111].

References

Gwyn Bellamy, Travis Schedler, and Ulrich Thiel, Hyperplane arrangements associated to symplectic quotient singularities, Phenomenological approach to algebraic geometry, Banach Center Publ., vol. 116, Polish Acad. Sci. Inst. Math., Warsaw, 2018, pp. 25–45. MR 3889149
Gwyn Bellamy and Ulrich Thiel, Highest weight theory for finite-dimensional graded algebras with triangular decomposition, Adv. Math. 330 (2018), 361–419. MR 3787549, DOI 10.1016/j.aim.2018.03.011
Roman Bezrukavnikov and Michael Finkelberg, Wreath Macdonald polynomials and the categorical McKay correspondence, Camb. J. Math. 2 (2014), no. 2, 163–190. With an appendix by Vadim Vologodsky. MR 3295916, DOI 10.4310/CJM.2014.v2.n2.a1
Ivan Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators, Internat. Math. Res. Notices 9 (1992), 171–180. MR 1185831, DOI 10.1155/S1073792892000199
Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI 10.2307/2118632
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348. MR 1881922, DOI 10.1007/s002220100171
H. O. Foulkes, A survey of some combinatorial aspects of symmetric functions, Permutations (Actes Colloq., Univ. René-Descartes, Paris, 1972), Gauthier-Villars, Paris, 1974, pp. 79–92. MR 0416934
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
Iain Gordon, Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (2003), no. 3, 321–336. MR 1960942, DOI 10.1112/S0024609303001978
Stephen Griffeth, Macdonald polynomials as characters of Cherednik algebra modules, Math. Z. 277 (2014), no. 1-2, 317–323. MR 3205775, DOI 10.1007/s00209-013-1257-3
Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919, DOI 10.1090/S0894-0347-01-00373-3
Mark Haiman, Notes on Macdonald polynomials and the geometry of Hilbert schemes, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 1–64. MR 2059359, DOI 10.1007/978-94-010-0524-1_{1}
Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39–111. MR 2051783
Randall R. Holmes and Daniel K. Nakano, Brauer-type reciprocity for a class of graded associative algebras, J. Algebra 144 (1991), no. 1, 117–126. MR 1136900, DOI 10.1016/0021-8693(91)90132-R
C. Kostka, Ueber den Zusammenhang zwischen einigen Formen von symmetrischen Functionen, J. Reine Angew. Math. 93 (1882), 89–123 (German). MR 1579917, DOI 10.1515/crll.1882.93.89
Alain Lascoux and Marcel-Paul Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 7, A323–A324 (French, with English summary). MR 472993
Nicholas A. Loehr and Jeffrey B. Remmel, A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin. 33 (2011), no. 2, 163–198. MR 2765321, DOI 10.1007/s10801-010-0238-4
I. G. Macdonald, A new class of symmetric functions, Actes du 20e Séminaire Lotharingien 372/S-20 (1988), 131–171.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Kendra Nelsen and Arun Ram, Kostka-Foulkes polynomials and Macdonald spherical functions, Surveys in combinatorics, 2003 (Bangor), London Math. Soc. Lecture Note Ser., vol. 307, Cambridge Univ. Press, Cambridge, 2003, pp. 325–370. MR 2011741
Stéphane Poirier, Cycle type and descent set in wreath products, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995), 1998, pp. 315–343. MR 1603753, DOI 10.1016/S0012-365X(97)00123-4
Wilhelm Specht, Eine Verallgemeinerung der Permutationsgruppen, Math. Z. 37 (1933), no. 1, 321–341 (German). MR 1545398, DOI 10.1007/BF01474578
Robert Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177. MR 372897, DOI 10.1016/0040-9383(75)90025-7
John R. Stembridge, On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), no. 2, 353–396. MR 1023791, DOI 10.2140/pjm.1989.140.353
U. Thiel, Champ: a Cherednik algebra Magma package, LMS J. Comput. Math. 18 (2015), no. 1, 266–307. MR 3361642, DOI 10.1112/S1461157015000054
Ulrich Thiel, Decomposition matrices are generically trivial, Int. Math. Res. Not. IMRN 7 (2016), 2157–2196. MR 3509951, DOI 10.1093/imrn/rnv204
Ulrich Thiel, Restricted rational Cherednik algebras, Representation theory—current trends and perspectives, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2017, pp. 681–745. MR 3644810
Ulrich Thiel, Blocks in flat families of finite-dimensional algebras, Pacific J. Math. 295 (2018), no. 1, 191–240. MR 3778331, DOI 10.2140/pjm.2018.295.191
Joshua Jeishing Wen, Wreath Macdonald polynomials as eigenstates, preprint arXiv:1904.05015v4, 2019.