Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group. (English) Zbl 1168.15017

Many works exist on various aspects of the distribution of the eigenvalues of large real symmetric or complex self-adjoint random matrices with simple stochastic structure. Recently, certain interest has been aroused in random matrices with more structure.
In this paper, the ensembles of highly structured random matrices defined in terms of representations of the symmetric group are investigated. The asymptotic behavior is analyzed as \(n \rightarrow \infty\) of the spectrum of the random matrices of the form
\[ \frac{1}{\sqrt{n-1}} \sum_{k=1}^{n-1}Z_{nk} \rho_n((k, k+1)), \]
where the random variables \(Z_{nk}\) are i.i.d. standard Gaussian and the matrices \(\rho_n((k,k+1))\) are obtained by applying an irreducible unitary representation \(\rho_n\) of the symmetric group of permutations on the set \(\{1,2, \dots, n \}\) to the transposition \((k,k+1)\) that interchanges \(k\) and \(k+1\) and satisfies the Coxeter relations. Irreducible representations of the symmetric group on \(\{1,2, \dots, n \}\) are indexed by a partition \(\lambda_n\) of \(n\).
The author establishes that if \(\lambda_{n,1}\geq \lambda_{n,2}\geq \dots \geq 0\) is a partition of \(n\) corresponding to \(\rho_n\), \(\mu_{n,1} \geq \mu_{n,2} \geq \dots \geq 0\) is the corresponding conjugate partition of \(n\), \(\lim_{n \rightarrow \infty} \frac{\lambda_{n,i}}{n}= \rho_i\) for each \(i \geq 1\), and \(\lim_{n \rightarrow \infty} \frac{\mu_{n,j}}{n}= q_j\) for each \(j \geq 1\), then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean \(\theta Z\) and variance \(1-\theta^2\), where \(\theta\) is the constant \(\sum_i p_i^2 - \sum_j q_j^2\) and \(Z\) is a standard Gaussian random variable with mean \(0\) and variance \(\frac{1}{n-1}\).


15B52 Random matrices (algebraic aspects)
20C30 Representations of finite symmetric groups
60F05 Central limit and other weak theorems
Full Text: DOI arXiv


[1] Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611-677. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. · Zbl 0949.60077
[2] Bryc, W., Dembo, A. and Jiang, T. (2006). Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 1-38. · Zbl 1094.15009 · doi:10.1214/009117905000000495
[3] Borodin, A., Okounkov, A. and Olshanski, G. (2000). Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 481-515 (electronic). JSTOR: · Zbl 0938.05061 · doi:10.1090/S0894-0347-00-00337-4
[4] Corteel, S., Goupil, A. and Schaeffer, G. (2004). Content evaluation and class symmetric functions. Adv. Math. 188 315-336. · Zbl 1059.05104 · doi:10.1016/j.aim.2003.09.010
[5] Diaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. ( N.S. ) 40 155-178 (electronic). · Zbl 1161.15302 · doi:10.1090/S0273-0979-03-00975-3
[6] Frobenius, F. G. (1900). Über die Charaktere der symmetrischen Gruppe. Sitz. Konig. Preuss. Akad. Wissen. 516-534. Collected in Gesammeltte Abhandlungen III (1968), 148-166. Springer, Heidelberg. · JFM 31.0129.02
[7] Fulton, W. and Harris, J. (1991). Representation Theory. Graduate Texts in Mathematics 129 . Springer, New York. · Zbl 0744.22001
[8] Fulton, W. (1997). Young Tableaux : With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts 35 . Cambridge Univ. Press, Cambridge. · Zbl 0878.14034
[9] Fulman, J. (2005). Stein’s method and Plancherel measure of the symmetric group. Trans. Amer. Math. Soc. 357 555-570 (electronic). · Zbl 1054.05099 · doi:10.1090/S0002-9947-04-03499-3
[10] Fulman, J. (2006). An inductive proof of the Berry-Esseen theorem for character ratios. Ann. Comb. 10 319-332. · Zbl 1106.05102 · doi:10.1007/s00026-006-0290-x
[11] Fulman, J. (2006). Martingales and character ratios. Trans. Amer. Math. Soc. 358 4533-4552 (electronic). · Zbl 1089.05078 · doi:10.1090/S0002-9947-06-03865-7
[12] Greene, C. (1992). A rational-function identity related to the Murnaghan-Nakayama formula for the characters of S n . J. Algebraic Combin. 1 235-255. · Zbl 0780.05057 · doi:10.1023/A:1022435901373
[13] Hammond, C. and Miller, S. J. (2005). Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theoret. Probab. 18 537-566. · Zbl 1086.15024 · doi:10.1007/s10959-005-3518-5
[14] Hora, A. (1998). Central limit theorem for the adjacency operators on the infinite symmetric group. Comm. Math. Phys. 195 405-416. · Zbl 1053.46522 · doi:10.1007/s002200050395
[15] Houdré, C., Pérez-Abreu, V. and Üstünel, A. S. (1994). Chaos Expansions , Multiple Wiener-Itô Integrals and Their Applications . CRC Press, Boca Raton, FL. · Zbl 0839.00028
[16] Ingram, R. E. (1950). Some characters of the symmetric group. Proc. Amer. Math. Soc. 1 358-369. · Zbl 0054.01103 · doi:10.2307/2032385
[17] Ivanov, V. and Olshanski, G. (2002). Kerov’s central limit theorem for the Plancherel measure on Young diagrams. In Symmetric Functions 2001: Surveys of Developments and Perspectives. NATO Sci. Ser. II Math. Phys. Chem. 74 93-151. Kluwer Academic, Dordrecht. · Zbl 1016.05073
[18] James, G. and Kerber, A. (1981). The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and Its Applications 16 . Addison-Wesley, Reading, MA. With a foreword by P. M. Cohn, with an introduction by Gilbert de B. Robinson. · Zbl 0491.20010
[19] Kerov, S. (1993). Gaussian limit for the Plancherel measure of the symmetric group. C. R. Acad. Sci. Paris Sér. I Math. 316 303-308. · Zbl 0793.43001
[20] Lassalle, M. (2005). Explicitation of characters of the symmetric group. C. R. Math. Acad. Sci. Paris 341 529-534. · Zbl 1081.20014 · doi:10.1016/j.crma.2005.09.016
[21] Littlewood, D. E. (2006). The Theory of Group Characters and Matrix Representations of Groups . Amer. Math. Soc., Providence, RI. Reprint of the second (1950) edition. · Zbl 0038.16504
[22] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials , 2nd ed. Oxford Mathematical Monographs . Clarendon, Oxford. With contributions by A. Zelevinsky, Oxford Science Publications. · Zbl 0824.05059
[23] Mehta, M. L. (2004). Random Matrices , 3rd ed. Pure and Applied Mathematics ( Amsterdam ) 142 . Elsevier/Academic Press, Amsterdam. · Zbl 1107.15019
[24] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Springer, Berlin. · Zbl 1099.60003
[25] Rutherford, D. E. (1948). Substitutional Analysis . Edinburgh Univ. Press, Edinburgh. · Zbl 0038.01602
[26] Sagan, B. E. (2001). The Symmetric Group : Representations , Combinatorial Algorithms , and Symmetric Functions , 2nd ed. Graduate Texts in Mathematics 203 . Springer, New York. · Zbl 0964.05070
[27] Simon, B. (1996). Representations of Finite and Compact Groups. Graduate Studies in Mathematics 10 . Amer. Math. Soc., Providence, RI. · Zbl 0840.22001
[28] Śniady, P. (2006). Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probab. Theory Related Fields 136 263-297. · Zbl 1104.46035 · doi:10.1007/s00440-005-0483-y
[29] Shao, Q.-M. and Su, Z.-G. (2006). The Berry-Esseen bound for character ratios. Proc. Amer. Math. Soc. 134 2153-2159 (electronic). · Zbl 1093.60014 · doi:10.1090/S0002-9939-05-08177-3
[30] Stanley, R. P. (1999). Enumerative combinatorics. Cambridge Studies in Advanced Mathematics 62 . Cambridge Univ. Press, Cambridge. · Zbl 0928.05001
[31] Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. ( 2 ) 67 325-327. JSTOR: · Zbl 0085.13203 · doi:10.2307/1970008
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