ALGEBRAIC COMBINATORICS

no. 3
The perfect matching association scheme
Srinivasan, Murali K.1
1 Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 400076, India
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 559-591.

We revisit the Bose–Mesner algebra of the perfect matching association scheme. Our main results are

  • An inductive algorithm, based on solving linear equations, to compute the eigenvalues of the orbital basis elements given the central characters of the symmetric groups.
  • Universal formulas, as content evaluations of symmetric functions, for the eigenvalues of fixed orbitals.
  • An inductive construction of an eigenvector (the so called first Gelfand–Tsetlin vector) in each eigenspace leading to a different inductive algorithm (not using central characters) for the eigenvalues of the orbital basis elements.
Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.104
Classification: 05E10, 05E05, 05E30
Keywords: perfect matching association scheme, content evaluation of symmetric functions, Gelfand–Tsetlin vectors.
Srinivasan, Murali K. 1

1 Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 400076, India
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Srinivasan, Murali K. The perfect matching association scheme. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 559-591. doi : 10.5802/alco.104. https://alco.centre-mersenne.org/articles/10.5802/alco.104/

[1] Aker, Kürşat; Can, Mahir Bilen Generators of the Hecke algebra of (S 2n ,B n ), Adv. Math., Volume 231 (2012) no. 5, pp. 2465-2483 | DOI | MR | Zbl

[2] Bannai, Eiichi; Ito, Tatsuro Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, California, 1984 | Zbl

[3] Bump, Daniel Lie Groups, 2nd edition, Grad. Texts Math., 225, Springer, New York, 2013 | MR | Zbl

[4] Can, Mahir Bilen; Özden, Şafak Corrigendum to “Generators of the Hecke algebra of (S 2n ,B n )” [Adv. Math. 231 (2012), no. 5, 2465–2483], Adv. Math., Volume 308 (2017), pp. 1337-1339 | Zbl

[5] Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo Representation Theory of the Symmetric Groups. The Okounkov–Vershik approach, Character formulas, and Partition Algebras, Camb. Stud. Adv. Math., 121, Cambridge University Press, Cambridge, 2010 | Zbl

[6] Corteel, Sylvie; Goupil, Alain; Schaeffer, Gilles Content evaluation and class symmetric functions, Adv. Math., Volume 188 (2004) no. 2, pp. 315-336 | DOI | MR | Zbl

[7] Diaconis, Persi; Greene, Curtis Applications of Murphy’s elements (1989) (http://statweb.stanford.edu/~cgates/PERSI/papers/EFSNSF335.pdf)

[8] Diaconis, Persi; Holmes, Susan P. Random walks on trees and matchings, Electron. J. Probab., Volume 7 (2002), Paper no. 6, 17 pages | MR | Zbl

[9] Garsia, Adriano Young’s seminormal representation and Murphy elements of S n (2003) (http://www.math.ucsd.edu/~garsia/somepapers/Youngseminormal.pdf)

[10] Godsil, Christopher; Meagher, Karen Erdős–Ko–Rado Theorems: Algebraic Approaches, Cambridge Studies in Advanced Mathematics, 149, Cambridge University Press, Cambridge, 2016 | Zbl

[11] Godsil, Christopher; Meagher, Karen An algebraic proof of the Erdős–Ko–Rado theorem for intersecting families of perfect matchings, Ars Math. Contemp., Volume 12 (2017) no. 2, pp. 205-217 | DOI | Zbl

[12] Hanlon, Philip J.; Stanley, Richard P.; Stembridge, John R. Some combinatorial aspects of the spectra of normally distribited random matrices, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) (Contemporary Mathematics), Volume 138, American Mathematical Society, Providence, RI, 1992, pp. 151-175 | DOI | Zbl

[13] James, Gordon; Kerber, Adalbert The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981 | MR | Zbl

[14] Jucys, Algimantas -A. A. Symmetric polynomials and the center of the symmetric group ring, Rep. Math. Phys., Volume 5 (1974), pp. 107-112 | DOI | MR | Zbl

[15] Ku, Cheng Yeaw; Wong, Kok Bin Eigenvalues of the matching derangement graph, J. Algebr. Comb., Volume 48 (2018) no. 4, pp. 627-646 | MR | Zbl

[16] Lindzey, Nathan Erdős–Ko–Rado for perfect matchings, Eur. J. Comb., Volume 65 (2017), pp. 130-142 | DOI | MR | Zbl

[17] Macdonald, Ian Grant Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1995 | Zbl

[18] Moran, Gadi The center of ℤ[S n+1 ] is the set of symmetric polynomials in n commuting transposition-sums, Trans. Am. Math. Soc., Volume 332 (1992) no. 1, pp. 167-180 | MR | Zbl

[19] Murphy, G. E. A new construction of Young’s seminormal representation of the symmetric groups, J. Algebra, Volume 69 (1981) no. 2, pp. 287-297 | DOI | MR | Zbl

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

[21] Muzychuk, Mikhail On association schemes of the symmetric group S 2n acting on partitions of type 2 n , Bayreuther Mathematische Schriften, Volume 47 (1994), pp. 151-164 | MR | Zbl

[22] Okounkov, Andrei; Vershik, Anatoliĭ M. A new approach to the representation theory of the symmetric groups. II, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. mat. Inst. Steklov. (POMI) 307 (2004), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, 57-98, 281; translation in J. Math. Sci. (New York), Volume 131 (2005), pp. 5471-5494 | MR

[23] Prasad, Amritanshu Representation Theory. A Combinatorial Viewpoint, Camb. Stud. Adv. Math., 147, Cambridge University Press, Delhi, 2015 | MR | Zbl

[24] Sagan, Bruce E. The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions, Grad. Texts Math, 203, Springer-Verlag, New York, 2001 | Zbl

[25] Saxl, Jan On multiplicity free permutation representations, Finite geometries and designs (Lond. Math. Soc. Lect. Note Ser.), Volume 49, Cambridge University Press, 1981, pp. 337-353 | DOI | MR | Zbl

[26] Srinivasan, Murali K. A Maple program for computing θ ^ 2μ 2λ (2018) (http://www.math.iitb.ac.in/~mks/papers/EigenMatch.pdf)

[27] Stanley, Richard P. Enumerative Combinatorics - Volume 2, Camb. Stud. Adv. Math., 62, Cambridge University Press, Cambridge, 1999 | Zbl

[28] Tout, Omar Structure coefficients of the Hecke algebra of (𝒮 2n ,ℬ n ), Electronic Journal of Combinatorics, Volume 21 (2014) no. 4, Paper no. Paper 4.35, 41 pages | MR | Zbl

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