48 research outputs found
Locally free representations of quivers over commutative Frobenius algebras
In this paper we investigate locally free representations of a quiver Q over
a commutative Frobenius algebra R by arithmetic Fourier transform. When the
base field is finite we prove that the number of isomorphism classes of
absolutely indecomposable locally free representations of fixed rank is
independent of the orientation of Q. We also prove that the number of
isomorphism classes of locally free absolutely indecomposable representations
of the preprojective algebra of Q over R equals the number of isomorphism
classes of locally free absolutely indecomposable representations of Q over
R[t]/(t^2). Using these results together with results of Geiss, Leclerc and
Schroer we give, when k is algebraically closed, a classification of pairs
(Q,R) such that the set of isomorphism classes of indecomposable locally free
representations of Q over R is finite. Finally, when the representation is free
of rank 1 at each vertex of Q, we study the function that counts the number of
isomorphism classes of absolutely indecomposable locally free representations
of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial
in q and their generating function is rational and satisfies a functional
equation.Comment: 44 page
Torus orbits on homogeneous varieties and Kac polynomials of quivers
In this paper we prove that the counting polynomials of certain torus orbits
in products of partial flag varieties coincides with the Kac polynomials of
supernova quivers, which arise in the study of the moduli spaces of certain
irregular meromorphic connections on trivial bundles over the projective line.
We also prove that these polynomials can be expressed as a specialization of
Tutte polynomials of certain graphs providing a combinatorial proof of the
non-negativity of their coefficients
Fourier transform from the symmetric square representation of and
Let be a connected reductive group over and let
be an algebraic representation of the dual
group . Assuming that and are defined over
, Braverman and Kazhdan defined an operator on the space
of complex valued functions on
. In this paper we are interested in the case where is
either or and is the symmetric square representation
of . We construct a natural -equivariant embedding
and an involutive operator
(Fourier transform) on the space of functions
that extends Braverman-Kazhdan's
operator