42,947 research outputs found
Degenerate two-boundary centralizer algebras
Diagram algebras (e.g. graded braid groups, Hecke algebras, 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 on tensor space of the
form . We define the degenerate two-boundary
braid algebra 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
and and modules and indexed by
rectangular partitions. For this setting, we define the degenerate extended
two-boundary Hecke algebra as a quotient of
, and show that a quotient of is
isomorphic to a large subalgebra of the centralizer. We further study the
representation theory of 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 is given by combinatorial
formulas.Comment: 45 pages, to appear in Pacific Journal of Mathematic
Geometry of Pipe Dream Complexes
In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v \u3e w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of the pipe dream complex of the identity element. We examine pipe dream complexes in the case a permutation is a product of commuting adjacent transpositions. Finally, we consider pattern avoidance results. For 132-avoiding permutations, the Rothe diagram forms a Young diagram. In the case a permutation w has exactly one 132-pattern, the associated pipe dream complex is an m-dimensional simplex, where m = n choose 2 − l(w) − 1 and l(w) is the length of w. In the case of exactly two 132 patterns, there are three possible configurations. We include generalizations of these cases
Compact Kac algebras and commuting squares
We consider commuting squares of finite dimensional von Neumann algebras
having the algebra of complex numbers in the lower left corner. Examples
include the vertex models, the spin models (in the sense of subfactor theory)
and the commuting squares associated to finite dimensional Kac algebras. To any
such commuting square we associate a compact Kac algebra and we compute the
corresponding subfactor and its standard invariant in terms of it.Comment: 14 pages, some minor change
Fundamental groupoids of k-graphs
k-graphs are higher-rank analogues of directed graphs which were first
developed to provide combinatorial models for operator algebras of
Cuntz-Krieger type. Here we develop a theory of the fundamental groupoid of a
k-graph, and relate it to the fundamental groupoid of an associated graph
called the 1-skeleton. We also explore the failure, in general, of k-graphs to
faithfully embed into their fundamental groupoids.Comment: 12 page
Symmetric pairs and associated commuting varieties
We obtain a series of new results on the problem of irreducibility of
commuting varieties associated with symmetric pairs or, in other words,
-graded simple Lie algebras. In particular, we present many examples of
reducible commuting varieties and show that the number of irreducible
components can be arbitrarily large.Comment: 18 page
Invariant subspaces of and preserving compatibility
Operators of multiplication by independent variables on the space of square
summable functions over the torus and its Hardy subspace are considered.
Invariant subspaces where the operators are compatible are described.Comment: 17 pages, 3 figure
Almost Commuting Orthogonal Matrices
We show that almost commuting real orthogonal matrices are uniformly close to
exactly commuting real orthogonal matrices. We prove the same for symplectic
unitary matrices. This is in contrast to the general complex case, where not
all pairs of almost commuting unitaries are close to commuting pairs. Our
techniques also yield results about almost normal matrices over the reals and
the quaternions.Comment: 13 pages, 3 figure
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