678 research outputs found
An explicit bijection between semistandard tableaux and non-elliptic sl_3 webs
The sl_3 spider is a diagrammatic category used to study the representation
theory of the quantum group U_q(sl_3). The morphisms in this category are
generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that
non-elliptic webs are indexed by semistandard Young tableaux. They establish
this bijection via a recursive growth algorithm. Recently, Tymoczko gave a
simple version of this bijection in the case that the tableaux are standard and
used it to study rotation and joins of webs. We build on Tymoczko's bijection
to give a simple and explicit algorithm for constructing all non-elliptic sl_3
webs
Chebyshev polynomials and the Frohman-Gelca formula
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the
Kauffman bracket skein module of the torus. This basis is especially useful
because the Jones-Kauffman product can be described via a very simple
Product-to-Sum formula. Presented in this work is a diagrammatic proof of this
formula, which emphasizes and demystifies the role played by Chebyshev
polynomials.Comment: 13 page
An Explicit Bijection Between Semistandard Tableaux and Non-Elliptic sl3 Webs
The sl3 spider is a diagrammatic category used to study the representation theory of the quantum group Uq(sl3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and joins of webs. We build on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all non-elliptic sl3 webs
The Bar-Natan skein module of the solid torus and the homology of (n,n) Springer varieties
This paper establishes an isomorphism between the Bar-Natan skein module of
the solid torus with a particular boundary curve system and the homology of the
(n,n) Springer variety. The results build on Khovanov's work with crossingless
matchings and the cohomology of the (n,n) Springer variety. We also give a
formula for comultiplication in the Bar-Natan skein module for this specific
three-manifold and boundary curve system.Comment: 17 pages, many figure
A topological constructions for all two-row Springer varieties
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification, Khovanov provides a topological construction of (m,m) Springer varieties. Here we extend his construction to all two-row Springer varieties. Using the combinatorial and diagrammatic properties of this construction we provide a particularly useful homology basis and construct the Springer representation using this basis. We also provide a skein-theoretic formulation of the representation in this case
Promoting REU participation from students in underrepresented groups
Research experiences for undergraduates (REUs) are an important component of undergraduate education. However, at the 2012 Trends in Undergraduate Research in the Mathematical Sciences conference, questions were raised about why many REU programs see few applications from students that are members of underrepresented groups. We examine the benefits of REUs and factors preventing or promoting participation in REUs
Springer Representations on the Khovanov Springer Varieties
Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H*(Xn). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H*(Xn) is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of Sn corresponding to the partition (n/2, n/2)
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