Czechoslovak Mathematical Journal, online first, 7 pp.

Further determinant identities related to classical root systems

Wenchang Chu

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

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", Università del Salento, Via Prov. Lecce-Arnesano, P. O. Box 193, Lecce 73100, Italia, e-mail: chu.wenchang@unisalento.it

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 2010): 05E05, 15A15

DOI: 10.21136/CMJ.2017.0265-16

Full text available as PDF.

References:
[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