Linear versus spin: representation theory of the symmetric groups

Matsumoto, Sho¹ ; Śniady, Piotr²

¹ Graduate School of Science and Engineering Kagoshima University 1-21-35 Korimoto Kagoshima Japan
² Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8 00-956 Warszawa Poland

Algebraic Combinatorics, Tome 3 (2020) no. 1, pp. 249-280.

Résumé

We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $\xi$ with analogous normalized linear characters evaluated on the double partition $D\left(\xi \right)$. We also relate some natural filtration on the usual (linear) Kerov–Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups in terms of counting maps.

Détail

Reçu le : 2018-11-23
Révisé le : 2019-07-08
Accepté le : 2019-08-01
Publié le : 2020-02-12

DOI : 10.5802/alco.92

Classification : 20C25, 20C30, 05E05
Mots-clés : Projective representations of the symmetric groups, linear representations of the symmetric groups, asymptotic representation theory, Stanley character formula

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Biane, Philippe Representations of symmetric groups and free probability, Adv. Math., Volume 138 (1998) no. 1, pp. 126-181 | Zbl | MR | DOI

[2] Biane, Philippe Characters of symmetric groups and free cumulants, Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001) (Lecture Notes in Math.), Volume 1815, Springer, Berlin, 2003, pp. 185-200 | Zbl | MR | DOI

[3] Biane, Philippe On the formula of Goulden and Rattan for Kerov polynomials, Sém. Lothar. Comb., Volume 55 (2005/07), B55d, 5 pages | Zbl | MR

[4] Czyżewska-Jankowska, Agnieszka; Śniady, Piotr Bijection between oriented maps and weighted non-oriented maps, Electron. J. Comb., Volume 24 (2017) no. 3, 3.7, 34 pages | Zbl | MR

[5] De Stavola, Dario Asymptotic results for Representation Theory (2018) (https://arxiv.org/abs/1805.04065)

[6] Dołęga, Maciej; Śniady, Piotr Gaussian fluctuations of Jack-deformed random Young diagrams, Probab. Theory Relat. Fields, Volume 174 (2019) no. 1-2, pp. 133-176 | MR | Zbl | DOI

[7] Dołęga, Maciej; Féray, Valentin; Śniady, Piotr Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations, Adv. Math., Volume 225 (2010) no. 1, pp. 81-120 | Zbl | MR | DOI

[8] Dołęga, Maciej; Féray, Valentin; Śniady, Piotr Jack polynomials and orientability generating series of maps, Sém. Lothar. Comb., Volume 70 (2014), B70j, 50 pages | MR | Zbl

[9] Dousse, Jehanne; Féray, Valentin Asymptotics for skew standard Young tableaux via bounds for characters (2017) (https://arxiv.org/abs/1710.05652)

[10] Féray, Valentin Stanley’s formula for characters of the symmetric group, Ann. Comb., Volume 13 (2010) no. 4, pp. 453-461 | Zbl | MR | DOI

[11] Féray, Valentin; Śniady, Piotr Asymptotics of characters of symmetric groups related to Stanley character formula, Ann. Math. (2), Volume 173 (2011) no. 2, pp. 887-906 | Zbl | MR | DOI

[12] Féray, Valentin; Śniady, Piotr Zonal polynomials via Stanley’s coordinates and free cumulants, J. Algebra, Volume 334 (2011) no. 1, pp. 338-373 | Zbl | MR | DOI

[13] Goulden, Ian P.; Rattan, Amarpreet An explicit form for Kerov’s character polynomials, Trans. Am. Math. Soc., Volume 359 (2007) no. 8, pp. 3669-3685 | DOI | Zbl | MR

[14] Hora, Akihito The limit shape problem for ensembles of Young diagrams, SpringerBriefs in Mathematical Physics, 17, Springer, Tokyo, 2016, ix+73 pages | Zbl | MR | DOI

[15] Ivanov, Vladimir Gaussian Limit for Projective Characters of Large Symmetric Groups, J. Math. Sci., New York, Volume 121 (2004) no. 3, pp. 2330-2344 | MR | DOI

[16] Ivanov, Vladimir Plancherel measure on shifted Young diagrams, Representation theory, dynamical systems, and asymptotic combinatorics (Amer. Math. Soc. Transl. Ser. 2), Volume 217, Amer. Math. Soc., Providence, RI, 2006, pp. 73-86 | Zbl | MR | DOI

[17] Kerov, Serguei; Olshanski, Grigori Polynomial functions on the set of Young diagrams, C. R. Acad. Sci. Paris Sér. I Math., Volume 319 (1994) no. 2, pp. 121-126 | Zbl | MR

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

[19] Lando, Sergei K.; Zvonkin, Alexander K. Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer-Verlag, Berlin, 2004 | Zbl | MR | DOI

[20] Linusson, Svante; Potka, Samu; Sulzberger, Robin On random shifted standard Young tableaux and $132$-avoiding sorting networks (2018) (https://arxiv.org/abs/1804.01795)

[21] Macdonald, Ian Grant 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) | Zbl | MR

[22] Matsumoto, Sho A spin analogue of Kerov polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 14 (2018), 053, 13 pages | Zbl | MR

[23] Matsumoto, Sho; Śniady, Piotr Stanley character formula for the spin characters of the symmetric groups, Séminaire Lotharingien de Combinatoire, Volume 82B (2019), #1, 12 pages Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana)

[24] Matsumoto, Sho; Śniady, Piotr Random strict partitions and random shifted tableaux, Selecta Mathematica, Volume 26 (2020) no. 1, 10, 59 pages | DOI

[25] Méliot, Pierre-Loïc Representation theory of symmetric groups, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2017, xvi+666 pages | Zbl | MR | DOI

[26] Nazarov, Maxim Young’s orthogonal form of irreducible projective representations of the symmetric group, J. Lond. Math. Soc., II. Ser., Volume 42 (1990) no. 3, pp. 437-451 | Zbl | MR | DOI

[27] Petrullo, Pasquale; Senato, Domenico Explicit formulae for Kerov polynomials, J. Algebr. Comb., Volume 33 (2011) no. 1, pp. 141-151 | Zbl | MR | DOI

[28] Rattan, Amarpreet; Śniady, Piotr Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula, Adv. Math., Volume 218 (2008) no. 3, pp. 673-695 | DOI | Zbl | MR

[29] Schur, Issai Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 139 (1911), pp. 155-250 | Zbl | MR | DOI

[30] Śniady, Piotr Gaussian fluctuations of characters of symmetric groups and of Young diagrams, Probab. Theory Relat. Fields, Volume 136 (2006) no. 2, pp. 263-297 | Zbl | MR | DOI

[31] Śniady, Piotr Combinatorics of asymptotic representation theory, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, pp. 531-545 | MR

[32] Śniady, Piotr Stanley character polynomials, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 323-334 | DOI | Zbl | MR

[33] Śniady, Piotr Asymptotics of Jack characters, J. Comb. Theory, Ser. A, Volume 166 (2019), pp. 91-143 | MR | Zbl | DOI

[34] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | Zbl | MR | DOI

[35] Stanley, Richard P. Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Comb., Volume 50 (2003/04), B50d, 11 pages | Zbl | MR

[36] Stanley, Richard P. A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group (2006) (https://arxiv.org/abs/math/0606467)

[37] Stembridge, John R. Shifted tableaux and the projective representations of symmetric groups, Adv. Math., Volume 74 (1989) no. 1, pp. 87-134 | Zbl | MR | DOI

[38] Wan, Jinkui; Wang, Weiqiang Lectures on spin representation theory of symmetric groups, Bull. Inst. Math., Acad. Sin. (N.S.), Volume 7 (2012) no. 1, pp. 91-164 | Zbl | MR