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A $q$-analogue of combinatorics concerning the Cartan matrix for theIwahori-Hecke algebra of type $A$ is investigated. We give several descriptionsfor the determinant of the graded Cartan matrix, which imply some combinatorialidentities. A conjectural expression for the elementary divisors is alsopresented.

Combinatorics for graded Cartan matrices of the Iwahori-Hecke algebra of type A