Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 981-987, 2017

Further determinant identities related to classical root systems

Wenchang Chu

Received May 27, 2016.   First published August 15, 2017.

Abstract:  By introducing polynomials in matrix entries, six determinants are evaluated which may be considered extensions of Vandermonde-like determinants related to the classical root systems.
Keywords:  Vandermonde determinant; symmetric function; classical root system
Classification MSC:  05E05, 15A15
DOI:  10.21136/CMJ.2017.0265-16

[1] G. Bhatnagar: A short proof of an identity of Sylvester. Int. J. Math. Math. Sci. 22 (1999), 431-435. DOI 10.1155/S0161171299224313 | MR 1695300 | Zbl 0929.01019
[2] W. Chu: Divided differences and symmetric functions. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 2 (1999), 609-618. MR 1719558 | Zbl 0935.05088
[3] W. Chu: Determinants and algebraic identities associated with the root systems of classical Lie algebras. Commun. Algebra 42 (2014), 3619-3633. DOI 10.1080/00927872.2013.790394 | MR 3196066 | Zbl 1291.05208
[4] W. Chu, L. V. Di Claudio: The Vandermonde determinant and generalizations associated with the classical Lie algebras. Ital. J. Pure Appl. Math. 20 (2006), 139-158. (In Italian.) MR 2247418 | Zbl 1150.15005
[5] F. J. Dyson: Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3 (1962), 140-156. DOI 10.1063/1.1703773 | MR 0143556 | Zbl 0105.41604
[6] W. Fulton, J. Harris: Representation Theory. Graduate Texts in Mathematics 129, Springer, New York (1991). DOI 10.1007/978-1-4612-0979-9 | MR 1153249 | Zbl 0744.22001
[7] I. J. Good: Short proof of a conjecture by Dyson. J. Math. Phys. 11 (1970), 1884. DOI 10.1063/1.1665339 | MR 0258644
[8] K. I. Gross, D. St. P. Richards: Constant term identities and hypergeometric functions on spaces of Hermitian matrices. J. Stat. Plann. Inference 34 (1993), 151-158. DOI 10.1016/0378-3758(93)90040-D | MR 1209996 | Zbl 0767.33010
[9] I. G. Macdonald: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, Clarendon Press, Oxford (1979). MR 0553598 | Zbl 0487.20007

Affiliations:   Wenchang Chu, School of Mathematics and Statistics, Zhoukou Normal University, Wenchang Road, Zhoukou 466001, Henan, People's Republic of China, and Dipartimento di Matematica e Fisica "Ennio De Giorgi", Universit√† del Salento, Via Prov. Lecce-Arnesano, P. O. Box 193, Lecce 73100, Italia, e-mail:

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