Additive Decompositions of Subgroups of Finite Fields

We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements and generalisations of results of C. Dartyge and A. Sarkozy on additive decompositions of quadratic residues and primitive roots modulo $p$. We use some new tools such the Karatsuba bound of double character sums and some results from additive combinatorics

Howe Duality and Combinatorial Character Formula for Orthosymplectic Lie superalgebras

We study the Howe dualities involving the reductive dual pairs $(O(d),spo(2m|2n))$ and $(Sp(d),osp(2m|2n))$ on the (super)symmetric tensor of \C^d\otimes\C^{m|n}. We obtain complete decompositions of this space with respect to their respective joint actions. We also use these dualities to derive a character formula for these irreducible representations of $spo(2m|2n)$ and $osp(2m|2n)$ that appear in these decompositions.Comment: 47 pages, LaTeX forma

Quarter-BPS states in orbifold sigma models with ADE singularities

We study the elliptic genera of two-dimensional orbifold CFTs, where the orbifolding procedure introduces du Val surface singularities on the target space. The N=4 character decompositions of the elliptic genus contributions from the twisted sectors at the singularities obey a consistent scaling property, and contain information about the arrangement of exceptional rational curves in the resolution. We also discuss how these twisted sector elliptic genera are related to twining genera and Hodge elliptic genera for sigma models with K3 target space.Comment: 13 pages + appendix. v2: minor changes, including additional reference

Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in a series of three dealing with groups of rank up to eight (and in particular, all exceptional Coxeter groups), we prove the conjecture for finite Coxeter groups of rank five and six, further developing the algorithmic tools described in the previous article. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik-Solomon characters of the groups considered.Comment: Final Version. 17 page

Mixed Tensors of the General Linear Supergroup

We describe the image of the canonical tensor functor from Deligne's interpolating category $Rep(GL_{m-n})$ to $Rep(GL(m|n))$ attached to the standard representation. This implies explicit tensor product decompositions between any two projective modules and any two Kostant modules of $GL(m|n)$, covering the decomposition between any two irreducible $GL(m|1)$-representations. We also obtain character and dimension formulas. For $m>n$ we classify the mixed tensors with non-vanishing superdimension. For $m=n$ we characterize the maximally atypical mixed tensors and show some applications regarding tensor products.Comment: v3: Improved exposition, corrected minor mistakes v2: shortened and revised version. Comments welcom