Expand all Collapse all | Results 1 - 13 of 13 |
1. CJM 2013 (vol 66 pp. 453)
A Remark on BMW algebra, $q$-Schur Algebras and Categorification We prove that the 2-variable BMW algebra
embeds into an algebra constructed from the HOMFLY-PT polynomial.
We also prove that the $\mathfrak{so}_{2N}$-BMW algebra embeds in the $q$-Schur algebra
of type $A$.
We use these results
to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$-BMW algebra.
Keywords:tangle algebras, BMW algebra, HOMFLY-PT Skein algebra, q-Schur algebra, categorification Categories:57M27, 81R50, 17B37, 16W99 |
2. CJM 2010 (vol 63 pp. 413)
Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating
functions for symmetric group characters are sometimes called
($S_n$) character immanants.
We point out a close connection between the identities of
Littlewood--Merris--Watkins
and Goulden--Jackson, which relate $S_n$ character immanants
to the determinant, the permanent and MacMahon's Master Theorem.
From these results we obtain a generalization
of Muir's identity.
Working with the quantum polynomial ring and the Hecke algebra
$H_n(q)$, we define quantum immanants that are generating
functions for Hecke algebra characters.
We then prove quantum analogs of the Littlewood--Merris--Watkins identities
and selected Goulden--Jackson identities
that relate $H_n(q)$ character immanants to
the quantum determinant, quantum permanent, and quantum Master Theorem
of Garoufalidis--L\^e--Zeilberger.
We also obtain a generalization of Zhang's quantization of Muir's
identity.
Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring Categories:15A15, 20C08, 81R50 |
3. CJM 2008 (vol 60 pp. 685)
Closed and Exact Functions in the Context of Ginzburg--Landau Models For a general vector field we exhibit two Hilbert spaces, namely
the space of so called \emph{closed functions} and the space of \emph{exact functions}
and we calculate the codimension of the space of exact functions
inside the larger space of closed functions.
In particular we provide a new approach for the known cases:
the Glauber field and the second-order Ginzburg--Landau field
and for the case of the fourth-order Ginzburg--Landau field.
Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries Categories:42B05, 81Q50, 42A16 |
4. CJM 2008 (vol 60 pp. 241)
Semi-Classical Wavefront Set and Fourier Integral Operators Here we define and prove some properties of the semi-classical
wavefront set. We also define and study semi-classical Fourier
integral operators and prove a generalization of Egorov's theorem to
manifolds of different dimensions.
Keywords:wavefront set, Fourier integral operators, Egorov theorem, semi-classical analysis Categories:35S30, 35A27, 58J40, 81Q20 |
5. CJM 2007 (vol 59 pp. 897)
The Ground State Problem for a Quantum Hamiltonian Model Describing Friction In this paper, we consider the quantum version of a Hamiltonian model
describing friction.
This model consists of
a particle which interacts with a bosonic reservoir representing a
homogeneous medium through which the particle moves. We show that if
the particle is confined, then the Hamiltonian admits a ground state
if and only if a suitable infrared condition is satisfied. The latter
is violated in the case of linear friction, but satisfied when the
friction force is proportional to a higher power of the particle
speed.
Categories:81Q10, 46N50 |
6. CJM 2001 (vol 53 pp. 756)
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions |
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions The proof of Lemma~3.4 in [F] relies on the incorrect equality $\mu_j
(AB) = \mu_j (BA)$ for singular values (for a counterexample, see [S,
p.~4]). Thus, Theorem~3.1 as stated has not been proven. However,
with minor changes, we can obtain a bound for the counting function in
terms of the growth of the Fourier transform of $|V|$.
Categories:47A10, 47A40, 81U05 |
7. CJM 2000 (vol 52 pp. 503)
The Level 2 and 3 Modular Invariants for the Orthogonal Algebras The `1-loop partition function' of a rational conformal field theory
is a sesquilinear combination of characters, invariant under a natural
action of $\SL_2(\bbZ)$, and obeying an integrality condition.
Classifying these is a clearly defined mathematical problem, and at
least for the affine Kac-Moody algebras tends to have interesting
solutions. This paper finds for each affine algebra $B_r^{(1)}$ and
$D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at
level 1 were classified. An extraordinary number of exceptionals
appear at level 2---the $B_r^{(1)}$, $D_r^{(1)}$ level 2
classification is easily the most anomalous one known and this
uniqueness is the primary motivation for this paper. The only level 3
exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$.
The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal
E}_6$-exceptional and $\E_8$-exceptional, respectively, in the
A-D-E classification for $A_1^{(1)}$, while the level 2 exceptionals
are related to the lattice invariants of affine~$u(1)$.
Keywords:Kac-Moody algebra, conformal field theory, modular invariants Categories:17B67, 81T40 |
8. CJM 1999 (vol 51 pp. 816)
A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type I consider a two-parameter family $B_{s,t}$ of unitary transforms
mapping an $L^{2}$-space over a Lie group of compact type onto a
holomorphic $L^{2}$-space over the complexified group. These were
studied using infinite-dimensional analysis in joint work with
B.~Driver, but are treated here by finite-dimensional means. These
transforms interpolate between two previously known transforms, and
all should be thought of as generalizations of the classical
Segal-Bargmann transform. I consider also the limiting cases $s
\rightarrow \infty$ and $s \rightarrow t/2$.
Categories:22E30, 81S30, 58G11 |
9. CJM 1998 (vol 50 pp. 1298)
Imprimitively generated Lie-algebraic Hamiltonians and separation of variables Turbiner's conjecture posits that a Lie-algebraic Hamiltonian
operator whose domain is a subset of the Euclidean plane admits a
separation of variables. A proof of this conjecture is given in
those cases where the generating Lie-algebra acts imprimitively.
The general form of the conjecture is false. A counter-example is
given based on the trigonometric Olshanetsky-Perelomov potential
corresponding to the $A_2$ root system.
Categories:35Q40, 53C30, 81R05 |
10. CJM 1998 (vol 50 pp. 756)
Estimates on renormalization group transformations We consider a specific realization of the renormalization group (RG)
transformation acting on functional measures for scalar quantum
fields which are expressible as a polymer expansion times an
ultra-violet cutoff Gaussian measure. The new and improved
definitions and estimates we present are sufficiently general and
powerful to allow iteration of the transformation, hence the
analysis of complete renormalization group flows, and hence the
construction of a variety of scalar quantum field theories.
Categories:81T08, 81T17 |
11. CJM 1998 (vol 50 pp. 538)
Upper bounds for the resonance counting function of SchrÃ¶dinger operators in odd dimensions The purpose of this note is to provide a simple proof of the sharp
polynomial upper bound for the resonance counting function of
a Schr\"odinger operator in odd dimensions. At the same time
we generalize the result to the class of super-exponentially
decreasing potentials.
Categories:47A10, 47A40, 81U05 |
12. CJM 1997 (vol 49 pp. 916)
Quantization of the $4$-dimensional nilpotent orbit of $\SL(3,\R)$ We give a new geometric model for the quantization
of the 4-dimensional conical (nilpotent) adjoint orbit $\OR$
of $\SL(3,\R)$. The space of quantization is the space of
holomorphic functions on ${\C}^2-\{0\})$ which are square integrable
with respect to a signed measure defined by a Meijer $G$-function.
We construct the quantization out a non-flat Kaehler structure on
${\C}^2-\{0\})$ (the universal cover of $\OR$) with Kaehler potential
$\rho=|z|^4$.
Categories:81S10, 32C17, 22E70 |
13. CJM 1997 (vol 49 pp. 160)
The Classical Limit of Dynamics for Spaces Quantized by an Action of ${\Bbb R}^{\lowercase{d}}$ We have previously shown how to construct a deformation quantization
of any locally compact space on which a vector group acts. Within this
framework we show here that, for a natural class of Hamiltonians, the
quantum evolutions will have the classical evolution as their
classical limit.
Categories:46L60, 46l55, 81S30 |