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Unitary $$t$$-groups. (English) Zbl 07257215
Summary: Relying on the main results of [GT], we classify all unitary $$t$$-groups for $$t\geq 2$$ in any dimension $$d\geq 2$$. We also show that there is essentially a unique unitary 4-group, which is also a unitary 5-group, but not a unitary $$t$$-group for any $$t\geq 6$$.

##### MSC:
 20C15 Ordinary representations and characters 05B30 Other designs, configurations 81P45 Quantum information, communication, networks (quantum-theoretic aspects)
##### Keywords:
unitary $$t$$-designs; unitary $$t$$-groups
CHEVIE; GAP
Full Text:
##### References:
 [1] [A] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math., 76 (1984), 469-514. · Zbl 0537.20023 [2] [Atlas] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford, 1985. · Zbl 0568.20001 [3] [FH] W. Fulton and J. Harris, Representation Theory, Grad. Texts in Math., 129, Springer-Verlag, New York, 1991. [4] [GAP] The GAP group, GAP — Groups, Algorithms, Programming, version 4.4, 2004, http://www.gap-system.org. [5] [GHMP] M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE — a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput., 7 (1996), 175-210. · Zbl 0847.20006 [6] [Gr] R. L. Griess, Jr., Automorphisms of extra special groups and nonvanishing degree 2 cohomology, Pacific J. Math., 48 (1973), 403-422. · Zbl 0283.20028 [7] [GAE] D. Gross, K. Audenaert and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys., 48 (2007), 052104. · Zbl 1144.81351 [8] [GMST] R. M. Guralnick, K. Magaard, J. Saxl and P. H. Tiep, Cross characteristic representations of symplectic and unitary groups, J. Algebra, 257 (2002), 291-347. · Zbl 1025.20002 [9] [GMT] R. M. Guralnick, K. Magaard and P. H. Tiep, Symmetric and alternating powers of Weil representations of finite symplectic groups, Bull. Inst. Math. Acad. Sin. (N.S.), 13 (2018), 443-461. · Zbl 07069218 [10] [GT] R. M. Guralnick and P. H. Tiep, Decompositions of small tensor powers and Larsen’s conjecture, Represent. Theory, 9 (2005), 138-208. · Zbl 1109.20040 [11] [He] C. Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. Algebra, 93 (1985), 151-164. · Zbl 0583.20003 [12] [Hu] B. Huppert, Endliche Gruppen. I, Springer-Verlag, 1967. [13] [Is] I. M. Isaacs, Character Theory of Finite Groups, AMS-Chelsea, Providence, 2006. · Zbl 1119.20005 [14] [L] M. W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3), 54 (1987), 477-516. · Zbl 0621.20001 [15] [Lu] F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math., 4 (2001), 135-169. · Zbl 1053.20008 [16] [Mag] K. Magaard, On the irreducibility of alternating powers and symmetric squares, Arch. Math., 63 (1994), 211-215. · Zbl 0824.20012 [17] [MM] K. Magaard and G. Malle, Irreducibility of alternating and symmetric squares, Manuscripta Math., 95 (1998), 169-180. · Zbl 0919.20009 [18] [MMT] K. Magaard, G. Malle and P. H. Tiep, Irreducibility of tensor squares, symmetric squares and alternating squares, Pacific J. Math., 202 (2002), 379-427. · Zbl 1072.20013 [19] [M] G. Malle, Almost irreducible tensor squares, Comm. Algebra, 27 (1999), 1033-1051. · Zbl 0931.20009 [20] [Mi] J. Michel, The development version of the CHEVIE package of GAP3, J. Algebra, 435 (2015), 308-336. [21] [RS] A. Roy and A. J. Scott, Unitary designs and codes, Des. Codes Cryptogr., 53 (2009), 13-31. · Zbl 1172.05310 [22] [S+] M. Schönert et al., GAP - Groups, Algorithms, and Programming, sixth edition, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 1997. [23] [Sc] A. J. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A, 41 (2008), 055308. · Zbl 1141.81009 [24] [ST] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math., 6 (1954), 274-304. · Zbl 0055.14305 [25] [TZ1] P. H. Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra, 24 (1996), 2093-2167. · Zbl 0901.20031 [26] [TZ2] P. H. Tiep and A. E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra, 192 (1997), 130-165. · Zbl 0877.20030 [27] [ZKGG] H. Zhu, R. Kueng, M. Grassl and D. Gross, The Clifford group fails gracefully to be a unitary 4-design, arXiv:1609.08172v1. [28] [Zs] K. · JFM 24.0176.02
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