Expand all Collapse all | Results 1 - 25 of 38 |
1. CJM Online first
Representation stability of power sets and square free polynomials The symmetric group $\mathcal{S}_n$ acts on the power
set $\mathcal{P}(n)$ and also on the set of
square free polynomials in $n$ variables. These
two related representations are analyzed from the stability point
of view. An application is given for the action of the symmetric
group on the cohomology of the pure braid group.
Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra Categories:20C30, 13A50, 20F36, 55R80 |
2. CJM 2014 (vol 66 pp. 902)
Corrigendum to Example in "Quantum Drinfeld Hecke Algebras" The last example of the article contains an error which we correct.
We also indicate some indices in Theorem 11.1 that were accidently transposed.
Keywords:quantum/skew polynomial rings, noncommutative Groebner bases Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 |
3. CJM Online first
On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or
on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.
Keywords:Littlewood polynomials, zeros, complex roots Categories:11R06, 11R09, 11C08 |
4. CJM 2013 (vol 66 pp. 525)
A Lift of the Schur and Hall-Littlewood Bases to Non-commutative Symmetric Functions We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions.
We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.
Keywords:Hall-Littlewood polynomial, symmetric function, quasisymmetric function, tableau Category:05E05 |
5. CJM 2013 (vol 66 pp. 874)
Quantum Drinfeld Hecke Algebras We consider finite groups acting on
quantum (or skew) polynomial rings. Deformations of the
semidirect product of the quantum polynomial ring with the acting group
extend symplectic reflection algebras and graded Hecke algebras
to the quantum setting over a field
of arbitrary characteristic.
We give necessary and sufficient conditions for such algebras to satisfy a
PoincarÃ©-Birkhoff-Witt property using the theory of noncommutative
GrÃ¶bner bases.
We include applications to the case of abelian groups
and the case of groups acting on coordinate rings of quantum planes.
In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology
gives an elegant description of the PBW conditions.
Keywords:skew polynomial rings, noncommutative GrÃ¶bner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 |
6. CJM 2013 (vol 66 pp. 625)
Classifying the Minimal Varieties of Polynomial Growth Let $\mathcal{V}$ be a variety of associative algebras generated by
an algebra with $1$ over a field of characteristic zero. This
paper is devoted to the classification of the varieties
$\mathcal{V}$ which are minimal of polynomial growth (i.e., their
sequence of codimensions growth like $n^k$ but any proper subvariety
grows like $n^t$ with $t\lt k$). These varieties are the building
blocks of general varieties of polynomial growth.
It turns out that for $k\le 4$ there are only a finite number of
varieties of polynomial growth $n^k$, but for each $k \gt 4$, the
number of minimal varieties is at least $|F|$, the cardinality of
the base field and we give a recipe of how to construct them.
Keywords:T-ideal, polynomial identity, codimension, polynomial growth, Categories:16R10, 16P90 |
7. CJM 2012 (vol 65 pp. 863)
Cumulants of the $q$-semicircular Law, Tutte Polynomials, and Heaps The $q$-semicircular distribution is a probability law that
interpolates between the Gaussian law and the semicircular law. There
is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free
cumulants one has to restrict the enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of
the classical cumulants. We show that like the free cumulants, they
are obtained by an enumeration of connected matchings, the weight
being now an evaluation of the Tutte polynomial of a so-called
crossing graph. The case $q=0$ of these cumulants was studied by
Lassalle using symmetric functions and hypergeometric series. We show
that the underlying combinatorics is explained through the theory of
heaps, which is Viennot's geometric interpretation of the
Cartier-Foata monoid. This method also gives a general formula for
the cumulants in terms of free cumulants.
Keywords:moments, cumulants, matchings, Tutte polynomials, heaps Categories:05A18, 05C31, 46L54 |
8. CJM 2012 (vol 65 pp. 600)
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials We establish asymptotics for Christoffel functions associated with
multivariate orthogonal polynomials. The underlying measures are assumed to
be regular on a suitable domain - in particular this is true if they are
positive a.e. on a compact set that admits analytic parametrization. As a
consequence, we obtain asymptotics for Christoffel functions for measures on
the ball and simplex, under far more general conditions than previously
known. As another consequence, we establish universality type limits in the
bulk in a variety of settings.
Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions Categories:42C05, 42C99, 42B05, 60B20 |
9. CJM 2012 (vol 65 pp. 171)
Optimal Polynomial Recurrence Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$.
We show, using Fourier analytic techniques, that if $N\geq
\exp\exp(C\varepsilon^{-1}\log\varepsilon^{-1})$ and
$A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that
\[\frac{|A\cap (A+P(n))|}{N}\gt \left(\frac{|A|}{N}\right)^2-\varepsilon.\]
In addition to this we also show, using the same Fourier analytic
methods, that if $A\subseteq\mathbb N$, then the set of
$\varepsilon$-optimal return times
\[R(A,P,\varepsilon)=\left\{n\in \mathbb N
\,:\,\delta(A\cap(A+P(n)))\gt \delta(A)^2-\varepsilon\right\}\]
is syndetic for every $\varepsilon\gt 0$. Moreover, we show that
$R(A,P,\varepsilon)$ is dense in every sufficiently long interval, in particular we show that
there exists an $L=L(\varepsilon,P,A)$ such that
\[\left|R(A,P,\varepsilon)\cap I\right|
\geq c(\varepsilon,P)|I|\]
for all intervals $I$ of natural numbers with $|I|\geq L$ and
$c(\varepsilon,P)=\exp\exp(-C\,\varepsilon^{-1}\log\varepsilon^{-1})$.
Keywords:Sarkozy, syndetic, polynomial return times Category:11B30 |
10. CJM 2012 (vol 64 pp. 318)
Cubic Polynomials with Periodic Cycles of a Specified Multiplier We consider cubic polynomials $f(z)=z^3+az+b$ defined over
$\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier
$\lambda$. In the case $N=1$, there are infinitely many such objects,
and in the case $N\geq 3$, only finitely many (subject to a mild
assumption). The case $N=2$ has particularly rich structure, and we
are able to describe all such cubic polynomials defined over the field
$\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.
Keywords:cubic polynomials, periodic points, holomorphic dynamics Category:37P35 |
11. CJM 2011 (vol 64 pp. 1036)
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields |
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields In this paper we study the extension problem, the
averaging problem, and the generalized ErdÅs-Falconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.
Keywords:extension problems, averaging operator, finite fields, ErdÅs-Falconer distance problems, homogeneous polynomial Categories:42B05, 11T24, 52C17 |
12. CJM 2011 (vol 64 pp. 481)
Some Functional Inequalities on Polynomial Volume Growth Lie Groups In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups.
We show in particular that improved Sobolev inequalities can be extended to this general framework
without the use of the Littlewood-Paley decomposition.
Keywords:Sobolev inequalities, polynomial volume growth Lie groups Category:22E30 |
13. 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 |
14. CJM 2010 (vol 63 pp. 200)
An Explicit Polynomial Expression for a $q$-Analogue of the 9-$j$ Symbols Using standard transformation and summation formulas for basic
hypergeometric series we obtain an explicit polynomial form of the
$q$-analogue of the 9-$j$ symbols, introduced by the author in a
recent publication. We also consider a limiting case in which the
9-$j$ symbol factors into two Hahn polynomials. The same
factorization occurs in another limit case of the corresponding
$q$-analogue.
Keywords:6-$j$ and 9-$j$ symbols, $q$-analogues, balanced and very-well-poised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$-Racah polynomials and their extensions Categories:33D45, 33D50 |
15. CJM 2010 (vol 63 pp. 181)
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the Askey--Wilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$-ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$-Jacobi polynomials is given.
Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients Categories:33D45, 42C05 |
16. CJM 2010 (vol 62 pp. 1058)
On a Conjecture of S. Stahl
S. Stahl conjectured that the zeros of genus polynomials are real. In
this note, we disprove this conjecture.
Keywords:genus polynomial, zeros, real Category:05C10 |
17. CJM 2010 (vol 62 pp. 261)
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials |
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials No abstract.
Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Problem Categories:34M10, 33C15, 33C47 |
18. CJM 2009 (vol 61 pp. 351)
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicity-free. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual Littlewood--Richardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials Categories:14L30, 22E46 |
19. CJM 2008 (vol 60 pp. 1240)
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The Bar-Natan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable Levine--Tristram signature.
Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism Categories:57M25, 57M27, 18G60 |
20. CJM 2008 (vol 60 pp. 958)
A Note on a Conjecture of S. Stahl S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.
Keywords:genus polynomial, zeros, real Categories:05C10, 05A15, 30C15, 26C10 |
21. 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 |
22. CJM 2007 (vol 59 pp. 1223)
CalderÃ³n--Zygmund Operators Associated to Ultraspherical Expansions We define the higher order Riesz transforms and the Littlewood--Paley
$g$-function
associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta
f'(\theta)+\l^2f(\theta)$. We prove that these operators are
Calder\'{o}n--Zygmund operators in the homogeneous type space
$((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted,
$H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.
Keywords:ultraspherical polynomials, CalderÃ³n--Zygmund operators Categories:42C05, 42C15frcs |
23. CJM 2007 (vol 59 pp. 186)
Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$,
arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer,
$h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and
$\alpha$ is a $K$-linear functional on the space $\krx$ of rational
functions in one variable $X$. Every pair $(h,\alpha)$ comes with a
polynomial $f$ in $K(X)[Y]$ called the regulator. When the module
${\mathcal M}$ admits non-trivial endomorphisms, $f$ must be linear or
quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and
only if $f$ is an irreducible quadratic. Then the $K$-algebra
$\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For
some height functions $h$ of infinite support $I$, the search for a
functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes
down to having functions $\eta\colon I\ar K$ such that no planar curve
intersects the graph of $\eta$ on a cofinite subset. If $K$ has
characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a
purely-simple Kronecker module ${\mathcal M}$ having non-trivial
endomorphisms, then $h$ attains the value $\infty$ at least once on
$\ktil$ and $h$ is finite-valued at least twice on
$\ktil$. Conversely all these $h$ form part of such triplets. The
proof of this result hinges on the fact that a rational function $r$
is a perfect square in $\krx$ if and only if $r$ is a perfect square
in the completions of $\krx$ with respect to all of its valuations.
Keywords:Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebra Categories:16S50, 15A27 |
24. CJM 2006 (vol 58 pp. 726)
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials |
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials We show that the value distribution (complex oscillation) of
solutions of certain periodic second order ordinary differential
equations studied by Bank, Laine and Langley is closely
related to confluent hypergeometric functions, Bessel functions
and Bessel polynomials. As a result, we give a complete
characterization of the zero-distribution in the sense of
Nevanlinna theory of the solutions for two classes of the ODEs.
Our approach uses special functions and their asymptotics. New
results concerning finiteness of the number of zeros
(finite-zeros) problem of Bessel and Coulomb wave functions with
respect to the parameters are also obtained as a consequence. We
demonstrate that the problem for the remaining class of ODEs not
covered by the above ``special function approach" can be
described by a classical Heine problem for differential
equations that admit polynomial solutions.
Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble Categories:34M10, 33C15, 33C47 |
25. CJM 2006 (vol 58 pp. 401)
On Pointwise Estimates of Positive Definite Functions With Given Support The following problem has been suggested by Paul Tur\' an. Let
$\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$
or in the torus $\TT^d$. Then, what is the largest possible value
of the integral of positive definite functions that are supported
in $\Omega$ and normalized with the value $1$ at the origin? From
this, Arestov, Berdysheva and Berens arrived at the analogous
pointwise extremal problem for intervals in $\RR$. That is, under
the same conditions and normalizations, the supremum of possible
function values at $z$ is to be found for any given point
$z\in\Omega$. However, it turns out that the problem for the real
line has already been solved by Boas and Kac, who gave several
proofs and also mentioned possible extensions to $\RR^d$ and to
non-convex domains as well.
Here we present another approach to the problem, giving the
solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we
elaborate on the fact that the problem is essentially
one-dimensional and investigate non-convex open domains as well.
We show that the extremal problems are equivalent to some more
familiar ones concerning trigonometric polynomials, and thus find
the extremal values for a few cases. An analysis of the
relationship between the problem for $\RR^d$ and that for $\TT^d$
is given, showing that the former case is just the limiting case
of the latter. Thus the hierarchy of difficulty is established, so
that extremal problems for trigonometric polynomials gain renewed
recognition.
Keywords:Fourier transform, positive definite functions and measures, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems Categories:42B10, 26D15, 42A82, 42A05 |