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Register a new userCrystals and Schur $P$-positive expansions
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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.
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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 particular, for the skew Schur function s_{delta_n / delta_{n-2}}, we discuss connections with Eulerian numbers and alternating permutations.
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 $mathbf{m}=(m_1, dots, m_r)$ such that $m_1+ dots + m_r =m$. We show that the Lie algebra $mathfrak{g}_{mathbf{Q}} (mathbf{m})$ is a filtered deformation of the current Lie algebra of $mathfrak{gl}_m$, and we can regard the algebra $mathcal{U}_{q, mathbf{Q}}(mathbf{m})$ as a $q$-analogue of $U(mathfrak{g}_{mathbf{Q}}(mathbf{m}))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $mathcal{U}_{q, mathbf{Q}}(mathbf{m})$ under a certain mild condition. We also study the representation theory for $mathfrak{g}_{mathbf{Q}}(mathbf{m})$ and $mathcal{U}_{q,mathbf{Q}}(mathbf{m})$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.
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.
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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