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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.
We prove Stanleys conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In p
We introduce a Lie algebra $mathfrak{g}_{mathbf{Q}}(mathbf{m})$ and an associative algebra $mathcal{U}_{q,mathbf{Q}}(mathbf{m})$ associated with the Cartan data of $mathfrak{gl}_m$ which is separated into $r$ parts with respect to $mathbf{m}=(m_1, do
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
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 t
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.