Séminaire Lotharingien de Combinatoire, 95B.3 (2026), 12 pp.
Ming Yean Lim
On Asymptotics of Unitriangular Groups
Abstract.
Enumerating irreducible representations of the group
UTn(q) of n × n upper unitriangular matrices over a
finite field of order q has proven to be a difficult problem. When
n is small, it has been computed that the number of irreducible
representations of UTn(q) is a polynomial
in q, but it
is not known if this holds true for all n. It is known however that
their degrees are powers of q. We investigate the number
Nn,e(q) of irreducible
representations of UTn(q) of degree
qe when e is large and show that their
asymptotics involve
counting 321-avoiding permutations with a given number of
inversions. We also propose a conjecture which entails similar results
for Nn,e(q) when e is
small, and also a conjecture on the
asymptotics of the number of conjugacy classes.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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