The Enumeration of own $t$-Dimensional Subspaces of a Space $V_{m}$ over the Field $GF(q)$

• Main Author: G. Egorychev
• Format: Article
• Language: English
• Published: Irkutsk State University 2016-09-01
• Series: Известия Иркутского государственного университета: Серия "Математика"
• Subjects: a number of subspaces of space; the method of coefficients; combinatorial sums
• Online Access: http://isu.ru/journal/downloadArticle?article=_af423ce4f0b745fe914f1da90b2557e9&lang=rus

In the Chevalley algebra over a field K associated with any system of roots, it is allocated the niltriangular subalgebra NΦ(K) with the basis $\{e_{r}(r\in \Phi ^{+}) \}$. In 2001 G.P. Egorychev and V.M. Levchuk had been put two problems of a enumeration of ideals: special ideals in the algebras of classical types (the problem 1) and all ideals (the problem 2). At their decision there is the problem of a finding of the number $V_{m,t}, 1< t< m$, all own t-dimensional subspaces of space $V_{m}$ over the field GF(q). Recently V.P. Krivokolesko and V.M. Levchuk have found an obvious expression for the number $V_{m,t}$ through a multiple sum from $q$-combinatorial numbers. Here by means of the method of coefficients of the calculation of combinatorial sums developed by the author in the late eighties, the integral representation for numbers $V_{m,t}$ is found. As consequence two simple computing formulas for these numbers were received.