# zbMATH — the first resource for mathematics

Random strict partitions and random shifted tableaux. (English) Zbl 07161024
Summary: We study asymptotics of random shifted Young diagrams which correspond to a given sequence of reducible projective representations of the symmetric groups. We show limit results (Law of Large Numbers and Central Limit Theorem) for their shapes, provided that the representation character ratios and their cumulants converge to zero at some prescribed speed. Our class of examples includes uniformly random shifted standard tableaux with prescribed shape as well as shifted tableaux generated by some natural combinatorial algorithms (such as shifted Robinson-Schensted-Knuth correspondence) applied to a random input.

##### MSC:
 20C25 Projective representations and multipliers 20C30 Representations of finite symmetric groups 60F05 Central limit and other weak theorems 05E10 Combinatorial aspects of representation theory
Full Text:
##### References:
 [1] Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Translated by N. Kemmer. Hafner Publishing Co., New York, pp. x+253 (1965) · Zbl 0135.33803 [2] Bernstein, D.; Henke, A.; Regev, A., Maximal projective degrees for strict partitions, Electron. J. Combin., 14, 1, 59 (2007) · Zbl 1158.05340 [3] Biane, P., Approximate factorization and concentration for characters of symmetric groups, Int. Math. Res. Not., 4, 179-192 (2001) · Zbl 1106.20304 [4] Biane, P., Representations of symmetric groups and free probability, Adv. Math., 138, 1, 126-181 (1998) · Zbl 0927.20008 [5] Borodin, A. M., Multiplicative central measures on the Schur graph, Journal of Mathematical Sciences, 96, 5, 3472-3477 (1999) · Zbl 1179.35038 [6] Brillinger, Dr, The calculation of cumulants via conditioning, Ann. Inst. Stat. Math., 21, 1, 215-218 (1969) · Zbl 0181.46103 [7] Dołȩga, M.; Féray, V.; Śniady, P., Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations, Adv. Math., 225, 1, 81-120 (2010) · Zbl 1231.05283 [8] Deya, A.; Nourdin, I., Convergence of Wigner integrals to the tetilla law, ALEA Lat. Am. J. Probab. Math. Stat., 9, 101-127 (2012) · Zbl 1285.46053 [9] Davis, R., Sagan, B.: Pattern-avoiding polytopes. Sém. Lothar. Combin. 78B, Art. 2, 12 (2017) [10] De Stavola, D.: Asymptotic results for representation theory (2017). arXiv:1805.04065 [math.CO] [11] Dołȩga, M.; Śniady, P., Gaussian fluctuations of Jackdeformed random Young diagrams, Probab. Theory Relat. ields, 174, 1-2, 133-176 (2019) · Zbl 1411.05281 [12] Fishel, S.; Nelson, L., Chains of maximum length in the Tamari lattice, Proc. Am. Math. Soc., 142, 10, 3343-3353 (2014) · Zbl 1326.06006 [13] Hoffman, Pn; Humphreys, Jf, Projective Representations of the Symmetric Groups. Oxford Mathematical Monographs. Q-functions and Shifted Tableaux, xiv+304 (1992), New York: Oxford Science Publications, New York [14] Hora, A., The Limit Shape Problem for Ensembles of Young Diagrams, ix+73 (2016), Tokyo: Springer, Tokyo · Zbl 1380.20001 [15] Ivanov, Vladimir; Olshanski, Grigori, Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams, Symmetric Functions 2001: Surveys of Developments and Perspectives, 93-151 (2002), Dordrecht: Springer Netherlands, Dordrecht · Zbl 1016.05073 [16] Ivanov, Vn, Gaussian limit for projective characters of large symmetric groups, J. Math. Sci., 121, 3, 2330-2344 (2004) · Zbl 1069.60025 [17] Ivanov, V.: Plancherel measure on shifted Young diagrams. In: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, vol. 217. Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc., Providence, RI, pp. 73-86 (2006). 10.1090/trans2/217/06 · Zbl 1111.05096 [18] Ivanov, Vn, Dimensions of skew-shifted young diagrams and projective characters of the infinite symmetric group, J. Math. Sci., 96, 5, 3517-3530 (1999) [19] Kerov, S. V., Transition probabilities for continual young diagrams and the Markov moment problem, Functional Analysis and Its Applications, 27, 2, 104-117 (1993) · Zbl 0808.05098 [20] Kerov, S., Gaussian limit for the Plancherel measure of the symmetric group, C. R. Acad. Sci. Paris Sér. I Math., 316, 4, 303-308 (1993) · Zbl 0793.43001 [21] Kerov, S.: Interlacing measures. In: Kirillov’s Seminar on Representation Theory, vol. 181. Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc., Providence, RI, pp. 35-83 (1998). 10.1090/trans2/181/02 · Zbl 0890.05074 [22] Kleshchev, A., Linear and Projective Representations of Symmetric Groups, xiv+277 (2005), Cambridge: Cambridge University Press, Cambridge · Zbl 1080.20011 [23] Kerov, S.; Olshanski, G., Polynomial functions on the set of Young diagrams, C. R. Acad. Sci. Paris Sér. I Math., 319.2, 121-126 (1994) · Zbl 0830.20028 [24] Linusson, S., Potka, S.: New properties of the Edelman-Greene bijection (2018). arXiv:1804.10034 [math.CO] · Zbl 1411.05283 [25] Linusson, S., Potka, S., Sulzberger, R.: On random shifted standard Young tableaux and 132-avoiding sorting networks (2018). arXiv:1804.01795 [math.CO] [26] Logan, Bf; Shepp, La, A variational problem for random Young tableaux, Adv. Math., 26, 2, 206-222 (1977) · Zbl 0363.62068 [27] Macdonald, Ig, Symmetric Functions and Hall Polynomials, x+475 (1995), , New York: Oxford Science Publications, , New York · Zbl 0824.05059 [28] Matsumoto, S.: A spin analogue of Kerov polynomials. In: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), vol. 14 (2018), p. 053. eprint: 1803.01121 · Zbl 1391.05267 [29] Méliot, P-L, Representation Theory of Symmetric Groups. Discrete Mathematics and its Applications (Boca Raton), xvi+666 (2017), Boca Raton: CRC Press, Boca Raton · Zbl 1418.20001 [30] Mingo, Ja; Speicher, R., Free Probability and Random Matrices. Fields Institute Monographs, xiv+336 (2017), New York: Springer, New York · Zbl 1387.60005 [31] Matsumoto, S., Śniady, P.: Linear versus spin: representation theory of the symmetric groups. To appear in Algebraic combinatorics (2018). arXiv:1811.10434 [math.CO] [32] Okounkov, A.: The uses of random partitions. In: Fourteenth International Congress on Mathematical Physics. Word Scientists, pp. 379-403 (2003) · Zbl 1120.05301 [33] Pittel, B.; Romik, D., Limit shapes for random square Young tableaux, Adv. Appl. Math., 38, 2, 164-209 (2007) · Zbl 1122.60009 [34] Proctor, Ra, Dynkin diagram classification of $$\lambda$$-minuscule Bruhat lattices and of d-complete posets, J. Algebr. Combin., 9, 1, 61-94 (1999) · Zbl 0920.06003 [35] Sagan, B., On selecting a random shifted Young tableau, J. Algorithms, 1, 3, 213-234 (1980) · Zbl 0468.05008 [36] Sagan, Be, Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A, 45.1, 62-103 (1987) · Zbl 0661.05010 [37] Schur, J., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 139, 155-250 (1911) · JFM 42.0154.02 [38] Schilling, A., Braid moves in commutation classes of the symmetric group, Eur. J. Combin, 62, 15-34 (2017) · Zbl 1358.05308 [39] Serrano, L., The shifted plactic monoid, Math. Z., 266, 2, 363-392 (2010) · Zbl 1231.05279 [40] Śniady, P.: Asymptotics of symmetric groups representations. Introduction to the habilitation thesis. University of Wrocław (2006) [41] Śniady, P., Gaussian fluctuations of characters of symmetric groups and of Young diagrams, Probab. Theory Relat. Fields, 136, 2, 263-297 (2006) · Zbl 1104.46035 [42] Śniady, P.: Structure coefficients for Jack characters: approximate factorization property (2016). arXiv:1603.04268v1 [math.CO] [43] Śniady, P., Asymptotics of Jack characters, J. Combin. Theory Ser. A, 166, 91-143 (2019) · Zbl 07074986 [44] Speed, Tp, Cumulants and partition lattices, Aust. J. Statist., 25, 2, 378-388 (1983) · Zbl 0538.60023 [45] Stembridge, Jr, Shifted tableaux and the projective representations of symmetric groups, Adv. Math., 74, 1, 87-134 (1989) · Zbl 0677.20012 [46] Stembridge, Jr, On the fully commutative elements of Coxeter groups, J. Algebr. Combin., 5, 4, 353-385 (1996) · Zbl 0864.20025 [47] Vershik, Anatoly M., Asymptotic Combinatorics and Algebraic Analysis, Proceedings of the International Congress of Mathematicians, 1384-1394 (1995), Basel: Birkhäuser Basel, Basel · Zbl 0843.05003 [48] Veršik, Am; Kerov, Sv, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR, 233, 6, 1024-1027 (1977) [49] Worley, D.R.: A theory of shifted Young tableaux. Thesis (Ph.D.)-Massachusetts Institute of Technology. ProQuest LLC, Ann Arbor, MI (1984) [50] Wan, J., Wang, W.: Lectures on spin representation theory of symmetric groups. In: Bull. Inst. Math. Acad. Sin. (N.S.) 7.1, pp. 91-164 (2012) · Zbl 1280.20013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.