Details
| Original language | English |
|---|---|
| Pages (from-to) | 163-177 |
| Number of pages | 15 |
| Journal | Journal of algebraic combinatorics |
| Volume | 21 |
| Issue number | 2 |
| Publication status | Published - Mar 2005 |
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 χRC 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 χRC. Taking r large yields the Smith normal form of the full character table of S n . Analogous results are then given for spin characters.
Keywords
- Character, Smith normal form, Spin character, Symmetric group
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Journal of algebraic combinatorics, Vol. 21, No. 2, 03.2005, p. 163-177.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Properties of some character tables related to the symmetric groups
AU - Bessenrodt, Christine
AU - Olsson, Jørn B.
AU - Stanley, Richard P.
N1 - Funding Information: ∗Partially supported by The Danish National Research Council. †Partially supported by NSF grant #DMS-9988459.
PY - 2005/3
Y1 - 2005/3
N2 - 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 χRC 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 χRC. Taking r large yields the Smith normal form of the full character table of S n . Analogous results are then given for spin characters.
AB - 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 χRC 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 χRC. Taking r large yields the Smith normal form of the full character table of S n . Analogous results are then given for spin characters.
KW - Character
KW - Smith normal form
KW - Spin character
KW - Symmetric group
UR - http://www.scopus.com/inward/record.url?scp=18644381938&partnerID=8YFLogxK
U2 - 10.1007/s10801-005-6906-0
DO - 10.1007/s10801-005-6906-0
M3 - Article
AN - SCOPUS:18644381938
VL - 21
SP - 163
EP - 177
JO - Journal of algebraic combinatorics
JF - Journal of algebraic combinatorics
SN - 0925-9899
IS - 2
ER -