2,803 research outputs found
Generating functions for moments of the quasi-nilpotent DT-operator
We prove a recursion formula for generating functions of certain
renormalizations of *-moments of the DT(\delta_0,1)-operator T, involving an
operation \odot on formal power series and a transformation E that converts
\odot to usual multiplication. This recursion formula is used to prove that all
of these generating functions are rational functions, and to find a few of them
explicitly.Comment: 16 page
Maximal increasing sequences in fillings of almost-moon polyominoes
It was proved by Rubey that the number of fillings with zeros and ones of a
given moon polyomino that do not contain a northeast chain of size depends
only on the set of columns of the polyomino, but not the shape of the
polyomino. Rubey's proof is an adaption of jeu de taquin and promotion for
arbitrary fillings of moon polyominoes. In this paper we present a bijective
proof for this result by considering fillings of almost-moon polyominoes, which
are moon polyominoes after removing one of the rows. Explicitly, we construct a
bijection which preserves the size of the largest northeast chains of the
fillings when two adjacent rows of the polyomino are exchanged. This bijection
also preserves the column sum of the fillings. We also present a bijection that
preserves the size of the largest northeast chains, the row sum and the column
sum if every row of the fillings has at most one 1.Comment: 18 page
Quasifinite representations of classical Lie subalgebras of W_{1+infty}
We show that there are precisely two, up to conjugation, anti-involutions
sigma_{\pm} of the algebra of differential operators on the circle preserving
the principal gradation. We classify the irreducible quasifinite highest weight
representations of the central extension \hat{D}^{\pm} of the Lie subalgebra of
this algebra fixed by - sigma_{\pm}, and find the unitary ones.
We realize them in terms of highest weight representations of the central
extension of the Lie algebra of infinite matrices with finitely many non-zero
diagonals over the truncated polynomial algebra C[u] / (u^{m+1}) and its
classical Lie subalgebras of B, C and D types. Character formulas for positive
primitive representations of \hat{D}^{\pm} (including all the unitary ones) are
obtained. We also realize a class of primitive representations of \hat{D}^{\pm}
in terms of free fields and establish a number of duality results between these
primitive representations and finite-dimensional irreducible representations of
finite-dimensional Lie groups and supergroups. We show that the vacuum module
V_c of \hat{D}^+ carries a vertex algebra structure and establish a
relationship between V_c for half-integral central charge c and W-algebras.Comment: Latex, 77 page
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