A centralizer analogue to the Farahat–Higman algebra
Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 337-360.

We define a family of algebras FH m which generalise the Farahat–Higman algebra introduced in [4] by replacing the role of the center of the group algebra of the symmetric groups with centraliser algebras of symmetric groups. These algebras have a basis indexed by marked cycle shapes, combinatorial objects which generalise proper integer partitions. We analyse properties of marked cycle shapes and of the algebras FH m , demonstrating that some of the former govern the latter. The main theorem of the paper proves that the algebra FH m is isomorphic to the tensor product of the degenerate affine Hecke algebra with the algebra of symmetric functions.

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DOI: 10.5802/alco.336
Classification: 05E15
Keywords: symmetric group, Farahat–Higman algebra, centralizer, degenerate affine Hecke algebra
Creedon, Samuel 1

1 Department of Mathematics Uppsala University Ångströmlaboratoriet, Lägerhyddsvägen 1 Box 480, 751 06 Uppsala Sweden
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Creedon, Samuel. A centralizer analogue to the Farahat–Higman algebra. Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 337-360. doi : 10.5802/alco.336. https://alco.centre-mersenne.org/articles/10.5802/alco.336/

[1] Creedon, Samuel; De Visscher, Maud Defining an affine partition algebra, Algebr. Represent. Theory, Volume 26 (2023) no. 6, pp. 2913-2965 | DOI | MR

[2] Danz, Susanne; Ellers, Harald; Murray, John The centralizer of a subgroup in a group algebra, Proc. Edinb. Math. Soc. (2), Volume 56 (2013) no. 1, pp. 49-56 | DOI | MR

[3] Deligne, P. La catégorie des représentations du groupe symétrique S t , lorsque t n’est pas un entier naturel, Algebraic groups and homogeneous spaces (Tata Inst. Fund. Res. Stud. Math.), Volume 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 209-273 | MR | Zbl

[4] Farahat, H. K.; Higman, G. The centres of symmetric group rings, Proc. Roy. Soc. London Ser. A, Volume 250 (1959), pp. 212-221 | DOI | MR

[5] Goulden, I. P.; Jackson, D. M. Symmetric functions and Macdonald’s result for top connexion coefficients in the symmetric group, J. Algebra, Volume 166 (1994) no. 2, pp. 364-378 | DOI | MR

[6] Harman, Nate; Snowden, Andrew Oligomorphic groups and tensor categories, 2022 | arXiv

[7] Ivanov, V.; Kerov, S. The algebra of conjugacy classes in symmetric groups, and partial permutations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 256 (1999), p. 95-120, 265 | DOI | MR

[8] Jucys, A.-A. A. Symmetric polynomials and the center of the symmetric group ring, Rep. Mathematical Phys., Volume 5 (1974) no. 1, pp. 107-112 | DOI | MR

[9] Kannan, Arun S.; Ryba, Christopher Stable Centres II: Finite Classical Groups, 2021 | arXiv

[10] Khovanov, Mikhail Heisenberg algebra and a graphical calculus, Fund. Math., Volume 225 (2014) no. 1, pp. 169-210 | DOI | MR

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

[12] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | DOI | MR

[13] Méliot, Pierre-Loïc Partial isomorphisms over finite fields, J. Algebraic Combin., Volume 40 (2014) no. 1, pp. 83-136 | DOI | MR

[14] Molev, A. I.; Olshanski, G. I. Degenerate affine Hecke algebras and centralizer construction for the symmetric groups, J. Algebra, Volume 237 (2001) no. 1, pp. 302-341 | DOI | MR | Zbl

[15] Murphy, G. E. The idempotents of the symmetric group and Nakayama’s conjecture, J. Algebra, Volume 81 (1983) no. 1, pp. 258-265 | DOI | MR

[16] Okounkov, A. Yu.; Vershik, A. M. A new approach to the representation theory of symmetric groups. II, J. Math. Sci. (N.Y.), Volume 131 (2005), pp. 5471-5494 | DOI

[17] Özden, Şafak Stability of the centers of the symplectic group rings [Sp 2n (q)], J. Algebra, Volume 572 (2021), pp. 263-296 | DOI | MR

[18] Ryba, Christopher Stable centres of wreath products, Algebr. Comb., Volume 6 (2023) no. 2, pp. 413-455 | MR

[19] Tysse, Jill; Wang, Weiqiang The centers of spin symmetric group algebras and Catalan numbers, J. Algebraic Combin., Volume 29 (2009) no. 2, pp. 175-193 | DOI | MR

[20] Wan, Jinkui; Wang, Weiqiang Stability of the centers of group algebras of GL n (q), Adv. Math., Volume 349 (2019), pp. 749-780 | DOI | MR

[21] Wang, Weiqiang The Farahat-Higman ring of wreath products and Hilbert schemes, Adv. Math., Volume 187 (2004) no. 2, pp. 417-446 | DOI | MR

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