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Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascouxs statement), in types B, C, and D in a stable range for t=1, as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K_{lambda,mu}(t). We also give a geometric interpretation.
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Let $p$ be any prime. Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. Let $phi$ and $psi$ be linear characters of $P_n$ and let $N$ be the normaliser of $P_n$ in $S_n$. In this article we show that the inductions of $phi$ and $psi$ to $S_n$ are equal if, and only if, $phi$ and $psi$ are $N$--conjugate. This is an analogue for symmetric groups of a result of Navarro for $p$-solvable groups.
For $G={rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(chi,g)$ in ${rm Irr}(G)times G$ with $chi(g) eq 0$ satisfies $P_{n,q}to 0$ as $ntoinfty$.
We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $mf{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.
The crystals for a finite-dimensional complex reductive Lie algebra $mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{mathfrak{g}}$, constructed using the Dynkin diagram of $mathfrak{g}$, and its combinatorial action on any $mathfrak{g}$-crystal via Sch{u}tzenberger involutions. We compare this action with that of the Berenstein-Kirillov group on Gelfand-Tsetlin patterns. Henriques and Kamnitzer define an action of $C_n=C_{mathfrak{gl}_n}$ on $n$-tensor products of $mathfrak{g}$-crystals, for any $mathfrak{g}$ as above. We discuss the crystal corresponding to the $mathfrak{gl}_n times mathfrak{gl}_m$-representation $Lambda^N(mathbb{C}^n otimes mathbb{C}^m),$ derive skew Howe duality on the crystal level and show that the two types of cactus group actions agree in this setting. A future application of this result is discussed in studying two families of maximal commutative subalgebras of the universal enveloping algebra, the shift of argument and Gaudin algebras, where an algebraically constructed monodromy action matches that of the cactus group.
We present a list of ``local axioms and an explicit combinatorial construction for the regular $B_2$-crystals (crystal graphs of highest weight integrable modules over $U_q(sp_4)$). Also a new combinatorial model for these crystals is developed.
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