9,040 research outputs found
Representations of Rational Cherednik algebras with minimal support and torus knots
We obtain several results about representations of rational Cherednik
algebras, and discuss their applications. Our first result is the
Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik
algebra modules with minimal support. Our second result is an explicit formula
for the character of an irreducible minimal support module in type A_{n-1} for
c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of
"hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young
diagram. We use this formula and the work of Calaque, Enriquez and Etingof to
give explicit formulas for the characters of the irreducible equivariant
D-modules on the nilpotent cone for SL_m. Our third result is the construction
of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes
the construction of the Koszul-BGG resolution by Berest-Etingof-Ginzburg and
Gordon, and the calculation of its homology in type A. We also show in type A
that the differentials in the Koszul-BGG complex are uniquely determined by the
condition that they are nonzero homomorphisms of modules over the Cherednik
algebra. Finally, our fourth result is the symmetry theorem, which identifies
the quasispherical components in the representations with minimal support over
the rational Cherednik algebras H_{m/n}(S_n) and H_{n/m}(S_m). In fact, we show
that the simple quotients of the corresponding quasispherical subalgebras are
isomorphic as filtered algebras. This symmetry has a natural interpretation in
terms of invariants of torus knots.Comment: 45 pages, latex; the new version contains a new subsection 3.4 on the
Cohen-Macaulay property of subspace arrangements and a strengthened version
of Theorem 1.
The homotopy type of the complement of a coordinate subspace arrangement
The homotopy type of the complement of a complex coordinate subspace
arrangement is studied by fathoming out the connection between its topological
and combinatorial structures. A family of arrangements for which the complement
is homotopy equivalent to a wedge of spheres is described. One consequence is
an application in commutative algebra: certain local rings are proved to be
Golod, that is, all Massey products in their homology vanish.Comment: 42 page
Rational homotopy type of subspace arrangements with a geometric lattice
Let A be a subspace arrangement with a geometric lattice such that codim(x) >
1 for every x in A. Using rational homotopy theory, we prove that the
complement M(A) is rationally elliptic if and only if the sum of the orthogonal
subspaces is a direct sum. The homotopy type of M(A) is also given: it is a
product of odd dimensional spheres. Finally, some other equivalent conditions
are given, such as Poincare duality. Those results give a complete description
of arrangements (with geometric lattice and with the codimension condition on
the subspaces) such that M(A) is rationally elliptic, and show that most
arrangements have an hyperbolic complement.Comment: 7 pages, to be published in Proceedings of the AM
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