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no. 6
Higher Specht polynomials under the diagonal action
Gillespie, Maria1
1 Colorado State University Department of Mathematics 1874 Campus Delivery Fort Collins, CO 80523 (USA)
Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1731-1750.

We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables x 1 ,...,x n and y 1 ,...,y n under the diagonal action of the symmetric group S n . This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring R n due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles.

As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia–Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.396
Classification: 05E10, 05E05, 05E40, 20C30
Keywords: Diagonal harmonics, diagonal coinvariants, representation theory of the symmetric group, Young tableaux, Young symmetrizers, Garsia-Haiman modules, polynomial rings

Gillespie, Maria 1

1 Colorado State University Department of Mathematics 1874 Campus Delivery Fort Collins, CO 80523 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Gillespie, Maria. Higher Specht polynomials under the diagonal action. Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1731-1750. doi : 10.5802/alco.396. https://alco.centre-mersenne.org/articles/10.5802/alco.396/

[1] Adin, Ron M.; Remmel, Jeffrey B.; Roichman, Yuval The combinatorics of the Garsia-Haiman modules for hook shapes, Electron. J. Combin., Volume 15 (2008) no. 1, Paper no. 38, 42 pages | DOI | MR | Zbl

[2] Alfano, Joseph A basis for the Y 1 subspace of diagonal harmonic polynomials, Discrete Math., Volume 193 (1998) no. 1-3, pp. 17-31 | DOI | MR | Zbl

[3] Allen, Edward E. A conjecture of Procesi and the straightening algorithm of Rota, Proc. Nat. Acad. Sci. U.S.A., Volume 89 (1992) no. 9, pp. 3980-3984 | DOI | MR | Zbl

[4] Allen, Edward E. The descent monomials and a basis for the diagonally symmetric polynomials, J. Algebraic Combin., Volume 3 (1994) no. 1, pp. 5-16 | DOI | MR | Zbl

[5] Allen, Edward E. Bitableaux bases for the diagonally invariant polynomial quotient rings, Adv. Math., Volume 130 (1997) no. 2, pp. 242-260 | DOI | MR | Zbl

[6] Ariki, Susumu; Terasoma, Tomohide; Yamada, Hiro-Fumi Higher Specht polynomials, Hiroshima Math. J., Volume 27 (1997) no. 1, pp. 177-188 | MR | Zbl

[7] Carlsson, Erik; Mellit, Anton A proof of the shuffle conjecture, J. Amer. Math. Soc., Volume 31 (2018) no. 3, pp. 661-697 | DOI | MR | Zbl

[8] Carlsson, Erik; Oblomkov, Alexei Affine Schubert calculus and double coinvariants, 2018 | arXiv

[9] De Concini, Corrado; Procesi, Claudio Symmetric functions, conjugacy classes and the flag variety, Invent. Math., Volume 64 (1981) no. 2, pp. 203-219 | DOI | MR | Zbl

[10] Fulton, William Young tableaux: with applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR

[11] Garsia, A. M.; Procesi, C. On certain graded S n -modules and the q-Kostka polynomials, Adv. Math., Volume 94 (1992) no. 1, pp. 82-138 | DOI | MR | Zbl

[12] Garsia, Adriano; Zabrocki, Mike A basis for the Diagonal Harmonic Alternants, 2022 | arXiv

[13] Gillespie, M.; Rhoades, B. Higher Specht bases for generalizations of the coinvariant ring, Ann. Comb., Volume 25 (2021) no. 1, pp. 51-77 | DOI | MR | Zbl

[14] Haglund, J.; Haiman, M.; Loehr, N.; Remmel, J. B.; Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., Volume 126 (2005) no. 2, pp. 195-232 | DOI | MR | Zbl

[15] Haglund, James; Rhoades, Brendon; Shimozono, Mark Ordered set partitions, generalized coinvariant algebras, and the delta conjecture, Adv. Math., Volume 329 (2018), pp. 851-915 | DOI | MR | Zbl

[16] Haiman, Mark Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc., Volume 14 (2001) no. 4, pp. 941-1006 | DOI | MR | Zbl

[17] Haiman, Mark Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math., Volume 149 (2002) no. 2, pp. 371-407 | DOI | MR | Zbl

[18] Lascoux, Alain; Schützenberger, Marcel-Paul Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B, Volume 286 (1978) no. 7, p. A323-A324 | MR | Zbl

[19] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979, viii+180 pages | MR

[20] Peel, M. H. Specht modules and symmetric groups, J. Algebra, Volume 36 (1975) no. 1, pp. 88-97 | DOI | MR | Zbl

[21] Sagan, Bruce E. The symmetric group: representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | DOI | MR

[22] Sage Developers, The SageMath (Version 9.0) (2020) https://www.sagemath.org

[23] Stanley, Richard P. Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 3, pp. 475-511 | DOI | MR | Zbl

[24] Stembridge, John R. Some particular entries of the two-parameter Kostka matrix, Proc. Amer. Math. Soc., Volume 121 (1994) no. 2, pp. 367-373 | DOI | MR | Zbl

[25] Tanisaki, Toshiyuki Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tohoku Math. J. (2), Volume 34 (1982) no. 4, pp. 575-585 | DOI | MR | Zbl

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