Commutative Schur rings over symmetric groups II: The case n = 6

Yıl 2016, Cilt: 3 Sayı: 2, 61 - 80, 15.05.2016

Amanda E. Francis Stephen P. Humphries

https://doi.org/10.13069/jacodesmath.79635

Öz

We determine the commutative Schur rings over $S_6$ that contain the sum of all the transpositions in
$S_6$. There are eight such types (up to conjugacy), of which four have the set of all the transpositions
as a principal set of the Schur ring.

Anahtar Kelimeler

Schur ring, Conjugacy class, Symmetric group, Group algebra

Kaynakça

• [1] W. Bosma, J. Cannon, Magma handbook, University of Sydney, 1993.
• [2] C. W. Curtis, Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. Vol. 15. American Mathematical Soc., 1999.
• [3] S. A. Evdokimov, I. N. Ponomarenko, On a family of Schur rings over a finite cyclic group (Russian), Algebra i Analiz. 13(3) (2001) 139–154; translation in St. Petersburg Math. J. 13(3) (2002) 441–451.
• [4] S. P. Humphries, Commutative Schur rings over symmetric groups, J. Algebraic Combin. 42(4) (2015) 971–997.
• [5] S. P. Humphries, K. W. Johnson, A. Misseldine, Commutative Schur rings of maximal dimension, Comm. Algebra. 43(12) (2015) 5298–5327.
• [6] K. H. Leung, S. H. Man, On Schur rings over cyclic groups, Israel J. Math. 106(1) (1998) 251–267.
• [7] K. H. Leung, S. H. Man, On Schur rings over cyclic groups, II, J. Algebra. 183(2) (1996) 273–285.
• [8] M. E. Muzychuk, On the structure of basic sets of Schur rings over cyclic groups, J. Algebra. 169(2) (1994) 655–678.
• [9] M. Muzychuk, I. Ponomarenko, Schur rings, European J. Combin. 30(6) (2009) 1526–1539.
• [10] I. Schur, Zur Theorie der einfach transitiven Permutationsgruppen, 1933.
• [11] H. Wielandt, Finite permutation groups, Academic Press, 2014.