Irreducible representations of braid groups via quantized enveloping algebras (Q1921936)

The paper contains an announcement of some of the results of the author's Ph.D. thesis. For any Lie algebra \(\mathfrak{g}\), there is a natural action of the associated braid group on any integrable \(\mathbb{U}(\mathfrak{g})\)-module, which permutes the weights according to the natural action of the Weyl group on weights [see \textit{G. Lusztig}, Introduction to quantum groups, Birkhäuser, Boston (1993; Zbl 0788.17010)]. The paper takes the case of \(\mathfrak{g}={\mathfrak{sl}}_{n+1}\) and shows that there is a combinatorially indexed family of irreducible representations of the standard braid groups \(B_i\), obtained from such a construction. Let \(\Phi_n\) denote the set of all weakly decreasing sequences of positive integers \(\mu_n=(\mu_{0,n},\mu_{1,n},\ldots,\mu_{n,n})\) of length \(n+1\), and let \(|\mu_n|\) denote the sum of the elements of the sequence \(\mu_n\). Define a partial order on \(\cup_n\Phi_n\) generated by \(\mu_{n-1}<\mu_n\), a comparison of elements of \(\Phi_{n-1}\) and \(\Phi_n\), precisely when \(\mu_{i+1,n}\leq\mu_{i,n-1}\leq\mu_{i,n}\) for all \(i\). An element of \(\Phi_n\) determines a dominant weight of \({\mathfrak{sl}}_{n+1}\) and let \(V(\mu_n)\) denote the associated simple \({\mathbb{U}}({\mathfrak{sl}}_{n+1})\)-module. A basis for \(V(\mu_n)\) is indexed by the set of all maximal chains of length \(n+1\), ending at \(\mu_n\), the so-called \textsl{Gelfand-Zetlin schemes}, the basis vectors being obtained from a highest weight by the application of lowering operators as encoded in the scheme [see \textit{K. Ueno, T. Takebayashi} and \textit{Y. Shibukawa}, Lett. Math. Phys. 18, 215-221 (1989; Zbl 0685.17004)]. Suppose that \(\mu_n\in\Phi_n\) has \(|\mu_n|=(n+1)p\) for some integer \(p\). Let \(\Sigma_i(\mu_n)\) denote the subset of \(\Phi_i\) consisting of elements of order \((i+1)p\) and less than \(\mu_n\) in the order. The space \(V(\mu_n)\) defines a \(B_{n+1}\)-module, which splits as a direct sum, possibly with multiplicity when considered as a \(B_{i+1}\)-module, for any \(i<n\). The main result is that when considering only the weight zero component, the \(B_{i+1}\)-modules so obtained are multiplicity-free, irreducible and indexed by elements of \(\Sigma_i(\mu_n)\). As a corollary, a particular two-parameter family of irreducible \(B_n\)-modules is constructed for \(n>2\), demonstrating that such modules exist with arbitrarily large dimension. Reviewer's comment: The construction employed in the paper would appear to be at least closely related to an algebraic analogue of a topological construction of braid group representations due to the reviewer [\textit{R. Lawrence}, Commun. Math. Phys. 135, 141-191 (1990; Zbl 0716.20022)].