Geodesic
($q,t$)-analogues and $GL_{n}({\mathbb{F}}_{q})$
Journal of Algebraic Combinatorics, Tome 31 (2010) no. 3, pp. 411-454.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We start with a $( q, t)$-generalization of a binomial coefficient. It can be viewed as a polynomial in $t$ that depends upon an integer $q$, with combinatorial interpretations when $q$ is a positive integer, and algebraic interpretations when $q$ is the order of a finite field. These $( q, t)$-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald's "$7 ^{ th }$ variation" of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of $GL _{ n}(\mathbb F _{ q})$ GL_n(mathbbF_q) .
Keywords: keywords $q$-binomial, $q$-multinomial, finite field, Gaussian coefficient, invariant theory, Coxeter complex, Tits building, Steinberg character, principal specialization 
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     title = {($q,t$)-analogues and $GL_{n}({\mathbb{F}}_{q})$},
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Reiner, Victor; Stanton, Dennis. ($q,t$)-analogues and $GL_{n}({\mathbb{F}}_{q})$. Journal of Algebraic Combinatorics, Tome 31 (2010) no. 3, pp. 411-454. http://geodesic.mathdoc.fr/item/JAC_2010__31_3_a1/