Abstract. We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied QSn-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.
The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an identification of the centers of the affine and degenerate affine BMW algebras in terms of rings of symmetric functions which satisfy a "cancellation property" or "wheel condition" (in the degenerate case, a reformulation of a result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as the cohomology and K-theory of isotropic Grassmanians and symplectic loop Grassmanians. We also establish new intertwiner-like identities which, when projected to the center, produce the recursions for central elements given previously by Nazarov for degenerate affine BMW algebras, and by Beliakova-Blanchet for affine BMW algebras.
We study the category F n of finite-dimensional integrable representations of the periplectic Lie superalgebra p(n). We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter 0 on the category F n by certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for p(n) resembling those for gl(m n). We discover two natural highest weight structures. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of irreducibles in standard and costandard modules and classify the blocks of F n . We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free.
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Diagram algebras (for example, graded braid groups, Hecke algebras and Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the action of a complex reductive Lie algebra g on tensor space of the form M ⊗ N ⊗ V ⊗k . We define the degenerate twoboundary braid algebra Ᏻ k and show that centralizer algebras contain quotients of this algebra in a general setting. As an example, we study in detail the combinatorics of special cases corresponding to Lie algebras gl n and sl n and modules M and N indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra Ᏼ ext k as a quotient of Ᏻ k , and show that a quotient of Ᏼ ext k is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of Ᏼ ext k to find that the seminormal representations are indexed by a known family of partitions. The bases for the resulting modules are given by paths in a lattice of partitions, and the action of Ᏼ ext k is given by combinatorial formulas.
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