# ALGEBRAIC COMBINATORICS

Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes
Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 23-45
Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of the combinatorial properties of the set partition combinatorics of the full uni-triangular groups, including combinatorial indexing sets, dimensions, and computable character formulas. Associated with these supercharacter theories is also a family of polytopes whose integer lattice points give the theories geometric underpinnings.
Accepted : 2017-08-18
Published online : 2018-01-29
DOI : https://doi.org/10.5802/alco.3
Classification:  05E10,  20C33
Keywords: supercharacters, integral polytopes, finite unipotent groups, unipotent radicals
@article{ALCO_2018__1_1_23_0,
author = {Thiem, Nathaniel},
title = {Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {1},
number = {1},
year = {2018},
pages = {23-45},
doi = {10.5802/alco.3},
zbl = {06882333},
mrnumber = {3857158},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_1_23_0}
}

Thiem, Nathaniel. Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 23-45. doi : 10.5802/alco.3. https://alco.centre-mersenne.org/item/ALCO_2018__1_1_23_0/

[1] Aguiar, M.; André, C.; Benedetti, C.; Bergeron, N.; Chen, Z.; Diaconis, P.; Hendrickson, A.; Hsiao, S.; Isaacs, I.M.; Jedwab, A.; Johnson, K.; Karaali, G.; Lauve, A.; Le, T.; Lewis, S.; Li, H.; Magaard, K.; Marberg, E.; Novelli, J-C.; Pang, A.; Saliola, F.; Tevlin, L.; Thibon, J-Y.; Thiem, N.; Venkateswaran, V.; Vinroot, C.R.; Yan, N.; Zabrocki, M. Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras, Adv. Math., Volume 229 (2012), pp. 2310-2337 | Article | MR 2880223 | Zbl 1237.05208

[2] Aguiar, M.; Bergeron, N.; Thiem, N. Hopf monoids from class functions on unitriangular matrices, Algebra Number Theory, Volume 7 (2013), pp. 1743-1779 | Article | MR 3117506 | Zbl 1276.05127

[3] André, C. Basic characters of the unitriangular group, J. Algebra, Volume 175 (1995), pp. 287-319 | Article | MR 1338979 | Zbl 0835.20052

[4] André, C.; Freitas, P.; Neto, A. A supercharacter theory for involutive algebra groups, J. Algebra, Volume 430 (2015), pp. 159-190 | Article | MR 3323980 | Zbl 1330.20006

[5] Andrews, S. Supercharacters of unipotent groups defined by involutions, J. Algebra, Volume 425 (2015), pp. 1-30 | Article | MR 3295975 | Zbl 1326.20012

[6] Arias-Castro, E.; Diaconis, P.; Stanley, R. A super-class walk on upper-triangular matrices, J. Algebra, Volume 278 (2004), pp. 739-765 | Article | MR 2071663 | Zbl 1056.60006

[7] Bergeron, N.; Thiem, N. A supercharacter table decomposition via power-sum symmetric functions, Internat. J. Algebra Comput., Volume 23 (2013), pp. 763-778 | Article | MR 3078055 | Zbl 1283.20004

[8] Brumbaugh, J.; Bulkow, M.; Fleming, P.; German, L. Garcia; Garcia, S.; Karaali, G.; Matt, M. Supercharacters, exponential sums, and the uncertainty principle, J. Number Theory, Volume 144 (2014), pp. 151-175 | Article | MR 3239156 | Zbl 1296.20004

[9] Burkett, S.; Lamar, J.; Lewis, M.; Wynn, C. Groups with exactly two supercharacter theories, Comm. Algebra, Volume 45 (2017), pp. 977-982 | Article | MR 3573353 | Zbl 1369.20005

[10] Chern, B.; Diaconis, P.; Kane, D.M.; Rhoades, R.C. Closed expressions for averages of set partition statistics, Res. Math. Sci., Volume 1 (2014), Art. 2, 32 pages | Article | MR 3338726 | Zbl 1339.15019

[11] De Loera, J.A.; Kim, E.D. Combinatorics and geometry of transportation polytopes: an update, Discrete geometry and algebraic combinatorics, Amer. Math. Soc., Providence, RI (Contemp. Math.) Volume 219 (2014), pp. 37-76 | MR 3289405 | Zbl 1360.90169

[12] Diaconis, P.; Isaacs, I.M. Supercharacters and superclasses for algebra groups, Trans. Amer. Math. Soc., Volume 360 (2008), pp. 2359-2392 | Article | MR 2373317 | Zbl 1137.20008

[13] Dipper, R.; Guo, Q. Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups, J. Pure Appl. Algebra, Volume 219 (2015), pp. 2559-2580 | Article | MR 3313496 | Zbl 1317.20004

[14] Fowler, C.; Garcia, S.; Karaali, G. Ramanujan sums as supercharacters, Ramanujan J., Volume 35 (2014), pp. 205-241 | Article | MR 3266478 | Zbl 1368.11090

[15] Marberg, E. A supercharacter analogue for normality, J. Algebra, Volume 332 (2011), pp. 334-365 | Article | MR 2774691 | Zbl 1243.20011

[16] Marberg, E.; Thiem, N. Superinduction for pattern groups, J. Algebra, Volume 321 (2009), pp. 3681-3703 | Article | MR 2517809 | Zbl 1190.20038

[17] Yan, N. Representations of finite unipotent linear groups by the method of clusters (Preprint)