Properties of Some Character Tables Related to the Symmetric Groups
Christine Bessenrodt
, Jørn B. Olsson2
and Richard P. Stanley2
2dagger
DOI: 10.1007/s10801-005-6906-0
Abstract
We determine invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups S n and their double covers. In particular, we give a simple computation, based on the theory of Hall-Littlewood symmetric functions, of the determinant of the regular character table 
RC of S n with respect to an integer r 
2. This result had earlier been proved by Olsson in a longer and more indirect manner. As a consequence, we obtain a new proof of the Mathas 
Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke algebra. When r is prime we determine the Smith normal form of RC. Taking r large yields the Smith normal form of the full character table of S n. Analogous results are then given for spin characters.
RC of S n with respect to an integer r 2. This result had earlier been proved by Olsson in a longer and more indirect manner. As a consequence, we obtain a new proof of the Mathas Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke algebra. When r is prime we determine the Smith normal form of RC. Taking r large yields the Smith normal form of the full character table of S n. Analogous results are then given for spin characters.Pages: 163–177
Keywords: keywords symmetric group; character; spin character; Smith normal form
Full Text: PDF
References
1. C. Bessenrodt and J.B. Olsson, “Spin representations and powers of 2,” Algebras and Representation Theory 3 (2000), 289-300.
2. C. Bessenrodt and J.B. Olsson, “A note on Cartan matrices for symmetric groups,” Arch. Math. 81 (2003), 497-504.
3. J. Brundan and A. Kleshchev, “Cartan determinants and Shapovalov forms,” Math. Ann. 324 (2002), 431-449.
4. S. Donkin, “Representations of Hecke algebras and characters of symmetric groups,” Studies in Memory of Issai Schur, Progress in Mathematics 210, Birkh\ddot auser Boston, 2003, pp. 158-170.
5. F.R. Gantmacher, The Theory of Matrices, vol. 1, Chelsea, New York, 1960.
6. J.W.L. Glaisher, “A theorem in partitions,” Messenger of Math. 12 (1883), 158-170.
7. P. Hoffman and J.F. Humphreys, Projective Representations of the Symmetric Groups, Oxford University Press, Oxford, 1992.
8. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, New York, 1981.
9. B. K\ddot ulshammer, J.B. Olsson, and G.R. Robinson, “Generalized blocks for symmetric groups,” Invent. Math. 151 (2003), 513-552.
10. I.G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995.
11. M. Newman, Integral Matrices, Academic Press, New York, 1972.
12. J.B. Olsson, “Regular character tables of symmetric groups,” The Electronic Journal of Combinatorics 10 (2003), N3.
13. I. Schur, “ \ddot Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen,” J. reine ang. Math. 39 (1911), 155-250, (Gesammelte Abhandlungen 1, pp. 346-441, Springer-Verlag, Berlin/New York, 1973).
14. R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, New York/Cambridge, 1999.
2. C. Bessenrodt and J.B. Olsson, “A note on Cartan matrices for symmetric groups,” Arch. Math. 81 (2003), 497-504.
3. J. Brundan and A. Kleshchev, “Cartan determinants and Shapovalov forms,” Math. Ann. 324 (2002), 431-449.
4. S. Donkin, “Representations of Hecke algebras and characters of symmetric groups,” Studies in Memory of Issai Schur, Progress in Mathematics 210, Birkh\ddot auser Boston, 2003, pp. 158-170.
5. F.R. Gantmacher, The Theory of Matrices, vol. 1, Chelsea, New York, 1960.
6. J.W.L. Glaisher, “A theorem in partitions,” Messenger of Math. 12 (1883), 158-170.
7. P. Hoffman and J.F. Humphreys, Projective Representations of the Symmetric Groups, Oxford University Press, Oxford, 1992.
8. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, New York, 1981.
9. B. K\ddot ulshammer, J.B. Olsson, and G.R. Robinson, “Generalized blocks for symmetric groups,” Invent. Math. 151 (2003), 513-552.
10. I.G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995.
11. M. Newman, Integral Matrices, Academic Press, New York, 1972.
12. J.B. Olsson, “Regular character tables of symmetric groups,” The Electronic Journal of Combinatorics 10 (2003), N3.
13. I. Schur, “ \ddot Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen,” J. reine ang. Math. 39 (1911), 155-250, (Gesammelte Abhandlungen 1, pp. 346-441, Springer-Verlag, Berlin/New York, 1973).
14. R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, New York/Cambridge, 1999.