86 research outputs found

### Critical classes, Kronecker products of spin characters, and the Saxl conjecture

Highlighting the use of critical classes, we consider constituents in
Kronecker products, in particular of spin characters of the double covers of
the symmetric and alternating groups. We apply results from the spin case to
find constituents in Kronecker products of characters of the symmetric groups.
Via this tool, we make progress on the Saxl conjecture; this claims that for a
triangular number $n$, the square of the irreducible character of the symmetric
group $S_n$ labelled by the staircase contains all irreducible characters of
$S_n$ as constituents. We find a large number of constituents in this square
which were not detected by other methods. Moreover, the investigation of
Kronecker products of spin characters inspires a spin variant of Saxl's
conjecture.Comment: 17 page

### Smith Normal Form of a Multivariate Matrix Associated with Partitions

Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and
Scoville to give a combinatorial interpretation of the entries of certain
matrices of determinant~1 in terms of lattice paths. Here we generalize this
result by refining the matrix entries to be multivariate polynomials, and by
determining not only the determinant but also the Smith normal form of these
matrices. A priori the Smith form need not exist but its existence follows from
the explicit computation. It will be more convenient for us to state our
results in terms of partitions rather than lattice paths.Comment: 12 pages; revised version (minor changes on first version); to appear
in J. Algebraic Combinatoric

### Cartan Invariants of Symmetric Groups and Iwahori-Hecke Algebras

K\"{u}lshammer, Olsson and Robinson conjectured that a certain set of numbers
determined the invariant factors of the $\ell$-Cartan matrix for $S_n$
(equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke
algebra $\mathcal{H}_n(q)$, where $q$ is a primitive $\ell$th root of unity).
We call these invariant factors Cartan invariants.
In a previous paper, the second author calculated these Cartan invariants
when $\ell=p^r$, $p$ prime, and $r\leq p$ and went on to conjecture that the
formulae should hold for all $r$. Another result was obtained, which is
surprising and counterintuitive from a block theoretic point of view. Namely,
given the prime decomposition $\ell=p_1^{r_1}... p_k^{r_k}$, the Cartan matrix
of an $\ell$-block of $S_n$ is a product of Cartan matrices associated to
$p_i^{r_i}$-blocks of $S_n$. In particular, the invariant factors of the Cartan
matrix associated to an $\ell$-block of $S_n$ can be recovered from the Cartan
matrices associated to the $p_i^{r_i}$-blocks.
In this paper, we formulate an explicit combinatorial determination of the
Cartan invariants of $S_n$--not only for the full Cartan matrix, \emph{but for
an individual block}. We collect evidence for this conjecture, by showing that
the formulae predict the correct determinant of the $\ell$-Cartan matrix. We
then go on to show that Hill's conjecture implies the conjecture of KOR

### Maximal multiplicative properties of partitions

Extending the partition function multiplicatively to a function on
partitions, we show that it has a unique maximum at an explicitly given
partition for any $n\neq 7$. The basis for this is an inequality for the
partition function which seems not to have been noticed before.Comment: 5 pages; in replacement: one typo in References corrected. To appear
in: Annals of Combinatoric

### Submatrices of character tables and basic sets

In this investigation of character tables of finite groups we study basic
sets and associated representation theoretic data for complementary sets of
conjugacy classes. For the symmetric groups we find unexpected properties of
characters on restricted sets of conjugacy classes, like beautiful
combinatorial determinant formulae for submatrices of the character table and
Cartan matrices with respect to basic sets; we observe that similar phenomena
occur for the transition matrices between power sum symmetric functions to
bounded partitions and the $k$-Schur functions introduced by Lapointe and
Morse. Arithmetic properties of the numbers occurring in this context are
studied via generating functions.Comment: 18 pages; examples added, typos removed, some further minor changes,
references update

### Residue symbols and Jantzen-Seitz partitions

Jantzen-Seitz partitions are those $p$-regular partitions of~$n$ which label
$p$-modular irreducible representations of the symmetric group $S_n$ which
remain irreducible when restricted to $S_{n-1}$; they have recently also been
found to be important for certain exactly solvable models in statistical
mechanics. In this article we study their combinatorial properties via a
detailed analysis of their residue symbols; in particular the $p$-cores of
Jantzen-Seitz partitions are determined

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