The Enumeration of own $t$-Dimensional Subspaces of a Space $V_{m}$ over the Field $GF(q)$
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 algebra...
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| Format: | Article |
| Language: | English |
| Published: |
Irkutsk State University
2016-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | http://isu.ru/journal/downloadArticle?article=_af423ce4f0b745fe914f1da90b2557e9&lang=rus |
| Summary: | 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. |
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| ISSN: | 1997-7670 2541-8785 |