ALGEBRAIC COMBINATORICS

no. 5
Comparing Fock spaces in types A (1) and A (2)
Fayers, Matthew1
1 Queen Mary University of London Mile End Road London E1 4NS U.K.
Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1347-1381.

We compare the canonical bases of level-1 quantised Fock spaces in affine types A (1) and A (2) , showing how to derive the canonical basis in type A 2n (2) from the the canonical basis in type A n (1) in certain weight spaces. In particular, we derive an explicit formula for the canonical basis in extremal weight spaces, which correspond to RoCK blocks of double covers of symmetric groups. In a forthcoming paper with Kleshchev and Morotti we will use this formula to find the decomposition numbers for RoCK blocks of double covers with abelian defect.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.300
Classification: 17B37, 05E10, 20C25, 20C30
Keywords: Quantum group, Fock space, symmetric groups, double covers, RoCK blocks
Fayers, Matthew 1

1 Queen Mary University of London Mile End Road London E1 4NS U.K.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Fayers, Matthew. Comparing Fock spaces in types $A^{(1)}$ and $A^{(2)}$. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1347-1381. doi : 10.5802/alco.300. https://alco.centre-mersenne.org/articles/10.5802/alco.300/

[1] Ariki, Susumu On the decomposition numbers of the Hecke algebra of G(m,1,n), J. Math. Kyoto Univ., Volume 36 (1996) no. 4, pp. 789-808 | MR | Zbl

[2] Brundan, Jonathan; Kleshchev, Alexander Odd Grassmannian bimodules and derived equivalences for spin symmetric groups, 2022 | arXiv

[3] Chuang, Joseph; Kessar, Radha Symmetric groups, wreath products, Morita equivalences, and Broué’s abelian defect group conjecture, Bull. London Math. Soc., Volume 34 (2002) no. 2, pp. 174-184 | DOI | Zbl

[4] Chuang, Joseph; Rouquier, Raphaël Derived equivalences for symmetric groups and 𝔰𝔩 2 -categorification, Ann. of Math. (2), Volume 167 (2008) no. 1, pp. 245-298 | DOI | MR | Zbl

[5] Chuang, Joseph; Tan, Kai Meng Some canonical basis vectors in the basic U q (𝔰𝔩 ^ n )-module, J. Algebra, Volume 248 (2002) no. 2, pp. 765-779 | DOI | Zbl

[6] Chuang, Joseph; Tan, Kai Meng Filtrations in Rouquier blocks of symmetric groups and Schur algebras, Proc. London Math. Soc. (3), Volume 86 (2003) no. 3, pp. 685-706 | DOI | MR | Zbl

[7] Ebert, Mark; Lauda, Aaron D.; Vera, Laurent Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications, 2022 | arXiv

[8] Fayers, Matthew Another runner removal theorem for v-decomposition numbers of Iwahori-Hecke algebras and q-Schur algebras, J. Algebra, Volume 310 (2007) no. 1, pp. 396-404 | DOI | MR | Zbl

[9] Fayers, Matthew The irreducible representations of the alternating group which remain irreducible in characteristic p, Trans. Amer. Math. Soc., Volume 368 (2016) no. 8, pp. 5807-5855 | DOI | MR | Zbl

[10] Fayers, Matthew Irreducible projective representations of the alternating group which remain irreducible in characteristic 2, Adv. Math., Volume 374 (2020), Paper no. 107340, 62 pages | MR | Zbl

[11] Fayers, Matthew Defect 2 spin blocks of symmetric groups and canonical basis coefficients, Represent. Theory, Volume 26 (2022), pp. 134-178 | DOI | MR | Zbl

[12] Fayers, Matthew; Kleshchev, Alexander; Morotti, Lucia Decomposition numbers for RoCK blocks of double covers of symmetric groups with abelian defect (in preparation)

[13] Hayashi, Takahiro Q-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras, Commun. Math. Phys., Volume 127 (1990), pp. 129-144 | DOI | MR | Zbl

[14] Humphreys, John F. Blocks of projective representations of the symmetric groups, J. London Math. Soc. (2), Volume 33 (1986) no. 3, pp. 441-452 | DOI | MR | Zbl

[15] Kessar, Radha Blocks and source algebras for the double covers of the symmetric and alternating groups, J. Algebra, Volume 186 (1996) no. 3, pp. 872-933 | DOI | MR | Zbl

[16] Kleshchev, Alexander Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, 163, Cambridge University Press, Cambridge, 2005, xiv+277 pages

[17] Kleshchev, Alexander; Livesey, Michael RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras, 2022 | arXiv

[18] Konvalinka, Matjaž; Lauve, Aaron Skew Pieri rules for Hall–Littlewood functions, J. Algebraic Combin., Volume 38 (2013) no. 3, pp. 499-518 | DOI | MR | Zbl

[19] Lascoux, Alain; Leclerc, Bernard; Thibon, Jean-Yves Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys., Volume 181 (1996) no. 1, pp. 205-263 | DOI | MR | Zbl

[20] Leclerc, Bernard; Miyachi, Hyohe Some closed formulas for canonical bases of Fock spaces, Represent. Theory, Volume 6 (2002), pp. 290-312 | DOI | MR | Zbl

[21] Leclerc, Bernard; Thibon, Jean-Yves q-deformed Fock spaces and modular representations of spin symmetric groups, J. Phys. A, Volume 30 (1997) no. 17, pp. 6163-6176 | DOI | MR | Zbl

[22] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications)

[23] Mathas, Andrew Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 15, American Mathematical Society, Providence, RI, 1999, xiv+188 pages

[24] Morris, A. O.; Yaseen, A. K. Some combinatorial results involving shifted Young diagrams, Math. Proc. Cambridge Philos. Soc., Volume 99 (1986) no. 1, pp. 23-31 | DOI | MR | Zbl

[25] Scopes, Joanna Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra, Volume 142 (1991) no. 2, pp. 441-455 | DOI | MR | Zbl

[26] Turner, W. Rock blocks, Mem. Amer. Math. Soc., Volume 202 (2009) no. 947, p. viii+102 | MR | Zbl

[27] Williamson, Geordie Schubert calculus and torsion explosion, J. Amer. Math. Soc., Volume 30 (2017) no. 4, pp. 1023-1046 (With a joint appendix with Alex Kontorovich and Peter J. McNamara) | DOI | MR | Zbl

[28] Yates, Dean A generalisation of bar-core partitions, Algebr. Comb., Volume 5 (2022) no. 4, pp. 667-698 | MR | Zbl

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