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This paper concerns the enumeration of simultaneous conjugacy classes of [Formula: see text]-tuples of commuting matrices in the upper triangular group [Formula: see text] and unitriangular group [Formula: see text] over the finite field [Formula: see text] of odd characteristic. This is done for [Formula: see text] and [Formula: see text], by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities [Formula: see text] for [Formula: see text] in each case.

Let G be an algebraic group. For d ≥ 1, we define the commuting prob-is the variety of commuting d-tuples in G. We prove that for a reductive group G when d is large, cp d (G) ∼ α n where n = dim(G), and α is the maximal dimension of an Abelian subgroup of G. For a finite reductive group G defined over the field Fq, we show that cp d+1 (G(Fq)) ∼ q (α−n)d , and give several examples.

This paper concerns the enumeration of simultaneous conjugacy classes of k-tuples of commuting matrices in the upper triangular group GTn(Fq) and unitriangular group U Tm(Fq) over the finite field Fq of odd characteristic. This is done for n = 2, 3, 4 and m = 3, 4, 5, by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities cp k for k ≤ 5 in each case.