Ribbon Schur operators (Q2462345)

This paper gives a new combinatorial approach to the ribbon tableau generating functions \({\mathcal G}_{\lambda/\mu}(X;q)\) (\(q\) a parameter), defined in terms of certain generalized Young tableaux, called ribbon tableaux, of shape \(\lambda/\mu\). These functions, introduced by \textit{A. Lascoux}, \textit{B. Leclerc}, and \textit{J.-Y. Thibon} [J. Math. Phys. 38, 1041--1068 (1997; Zbl 0869.05068)], are \(q\)-analogues of products of Schur functions, and generalize the Hall-Littlewood functions. Let \(\mathbf F\) denote a vector space over \(\mathbb C(q)\) spanned by a countable basis \(\{\lambda\mid \lambda\in\mathcal P\}\), indexed by partitions \(\lambda\). The author defines, for \(i\in\mathbb Z\), linear operators \(u_{i}^{(n)}:\mathbf F\rightarrow\mathbf F\), called ribbon Schur operators, which add ribbons (skew Young diagrams containing no \(2\times 2\) square) to partitions. Following \textit{S. Fomin} and \textit{C. Greene} [Discrete Math. 193, 179--200 (1998; Zbl 1011.05062)], he then studies non-commutative (skew) Schur functions \(s_{\lambda/\mu}(\underline{u})\) in these operators. The motivating question for this study is whether the \(q\)-Littlewood-Richardson coefficients \(c_{\lambda/\mu}^{\nu}(q)\in\mathbb N(q)\), defined by \({\mathcal G}_{\lambda/\mu}(X;q)= \sum_{\nu}c_{\lambda/\mu}^{\nu}(q) s_{\nu}(X)\), are non-negative polynomials (in \(q\)) for all skew shapes \(\lambda/\mu\). This condition is shown to be equivalent to the non-commutative Schur function \(s_{\nu}(\underline{u})\) being a non-negative sum of monomials in the \(u_{i}\) (Proposition 14). The question is formulated as Conjecture 15, which is then proved for \(\nu=(a,1^{b})\) (Theorem 17) and \(\nu=(s,2)\) (Theorem 17); the latter also gives the result for \(\nu=(2,2,1^{a})\) (by duality), and the author speculates that Theorems 17 and 19 may be combined to give an expression for \(\nu=(a,2,1^{b})\). The author's approach also yields new proofs of the symmetry of \({\mathcal G}_{\lambda/\mu}(X;q)\) (Corollary 13), a Cauchy identity (Theorem 6) and the action of the Heisenberg algebra on the Fock space \(\mathbf F\) of \(U_{q}(\widehat{\mathfrak{sl}_{n}})\) (Corollary 8); Cor. 8 implies Pieri and Cauchy formulae for \({\mathcal G}_{\lambda/\mu}(X;q)\). There are several concluding remarks (Sect. 8), for example on the relation between this work and affine Hecke algebras and canonical bases.