No CrossRef data available.
Article contents
A GEOMETRIC APPROACH TO POLYNOMIAL PROPERTY FOR PATTERN SUBGROUPS IN 
${{\mathrm {\mathbf {U}}}}_{\boldsymbol {a,b,c,d}}$
Part of:
Linear algebraic groups and related topics
Projective and enumerative geometry
Algebraic combinatorics
Representation theory of groups
Published online by Cambridge University Press: 19 November 2025
Abstract
Let 
${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and 
${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of 
$\mathrm {GL}_{n}$, where 
$a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of 
${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree 
$q^e$ for all these pattern groups. For pattern subgroups 
$G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by Oded Yacobi
References
Borel, A., Linear Algebraic Groups, Graduate Texts in Mathematics, 126 (Springer Science & Business Media, New York, 1991).10.1007/978-1-4612-0941-6CrossRefGoogle Scholar
Deodhar, V., ‘On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells’, Invent. Math. 79 (1985), 499–511.10.1007/BF01388520CrossRefGoogle Scholar
Diaconis, P. and Thiem, N., ‘Supercharacter formulas for pattern groups’, Trans. Amer. Math. Soc. 361 (2009), 3501–3533.10.1090/S0002-9947-09-04521-8CrossRefGoogle Scholar
Gonciulea, N. and Miller, C., ‘Mixed ladder determinantal varieties’, J. Algebra 231 (2000), 104–137.10.1006/jabr.2000.8358CrossRefGoogle Scholar
Gorla, E., ‘Mixed ladder determinantal varieties from two-sided ladders’, J. Pure Appl. Algebra 211 (2007), 433–444.10.1016/j.jpaa.2007.01.016CrossRefGoogle Scholar
Halasi, Z. and Pálfy, P., ‘The number of conjugacy classes in pattern groups is not a polynomial function’, J. Group Theory 14 (2011), 841–854.10.1515/jgt.2010.081CrossRefGoogle Scholar
Higman, G., ‘Enumerating p-groups. I: inequalities’, Proc. Lond. Math. Soc. (3) s3-10(1) (1960), 24–30.CrossRefGoogle Scholar
Humphreys, J., Linear Algebraic Groups, Graduate Texts in Mathematics, 21 (Springer Science & Business Media, New York, 1975).10.1007/978-1-4684-9443-3CrossRefGoogle Scholar
Isaacs, I. M., ‘Characters of groups associated with finite algebras’, J. Algebra 177 (1995), 708–730.10.1006/jabr.1995.1325CrossRefGoogle Scholar
Isaacs, I. M., ‘Counting characters of upper triangular groups’, J. Algebra 315 (2007), 689–719.10.1016/j.jalgebra.2007.01.027CrossRefGoogle Scholar
Kazhdan, D. and Lusztig, G., ‘Representations of Coxeter groups and Hecke algebras’, Invent. Math. 53 (1979), 165–184.10.1007/BF01390031CrossRefGoogle Scholar
Kirillov, A., ‘Merits and demerits of the orbit method’, Bull. Amer. Math. Soc. 36 (1999), 433–488.10.1090/S0273-0979-99-00849-6CrossRefGoogle Scholar
Lehrer, G., ‘Discrete series and the unipotent subgroup’, Compos. Math. 28 (1974), 9–19.Google Scholar
Nien, C., ‘Characters of 2-layered Heisenberg groups’, Linear Multilinear Algebra 70 (2022), 3633–3642.10.1080/03081087.2020.1849005CrossRefGoogle Scholar
Nien, C., ‘Characters of unipotent radicals of standard parabolic subgroups with 3 parts’, J. Algebraic Combin. 55 (2022), 325–333.10.1007/s10801-021-01052-8CrossRefGoogle Scholar
Serre, J., Linear Representations of Finite Groups, Graduate Texts in Mathematics, 42 (Springer Science & Business Media, New York, 1977).10.1007/978-1-4684-9458-7CrossRefGoogle Scholar