86 research outputs found

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

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    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 nn, the square of the irreducible character of the symmetric group SnS_n labelled by the staircase contains all irreducible characters of SnS_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

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    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

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    K\"{u}lshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the ℓ\ell-Cartan matrix for SnS_n (equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke algebra Hn(q)\mathcal{H}_n(q), where qq is a primitive ℓ\ellth root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when ℓ=pr\ell=p^r, pp prime, and r≤pr\leq p and went on to conjecture that the formulae should hold for all rr. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition ℓ=p1r1...pkrk\ell=p_1^{r_1}... p_k^{r_k}, the Cartan matrix of an ℓ\ell-block of SnS_n is a product of Cartan matrices associated to pirip_i^{r_i}-blocks of SnS_n. In particular, the invariant factors of the Cartan matrix associated to an ℓ\ell-block of SnS_n can be recovered from the Cartan matrices associated to the pirip_i^{r_i}-blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of SnS_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 ℓ\ell-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of KOR

    Maximal multiplicative properties of partitions

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    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≠7n\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

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    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 kk-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

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    Jantzen-Seitz partitions are those pp-regular partitions of~nn which label pp-modular irreducible representations of the symmetric group SnS_n which remain irreducible when restricted to Sn−1S_{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 pp-cores of Jantzen-Seitz partitions are determined
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