Expand all Collapse all | Results 1 - 22 of 22 |
1. CJM 2012 (vol 66 pp. 3)
On Hilbert Covariants Let $F$ denote a binary form of order $d$ over the
complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant
$\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an
order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give
defining equations for the image variety $X$ of an embedding $\mathbf{P}^r
\hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of
the Hilbert covariant; and simultaneously situate it into a wider class of
covariants called the GÃ¶ttingen covariants, all of which vanish on
$X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$
defines $X$ as a scheme. Finally, we exhibit a generalisation of the
GÃ¶ttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.
Keywords:binary forms, covariants, $SL_2$-representations Categories:14L30, 13A50 |
2. CJM 2012 (vol 64 pp. 1222)
Normality of Maximal Orbit Closures for Euclidean Quivers Let $\Delta$ be an Euclidean quiver. We prove that the closures of
the maximal orbits in the varieties of representations of $\Delta$
are normal and Cohen--Macaulay (even complete intersections).
Moreover, we give a generalization of this result for the tame
concealed-canonical algebras.
Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebra Categories:16G20, 14L30 |
3. CJM 2011 (vol 64 pp. 805)
Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 |
4. CJM 2011 (vol 64 pp. 409)
Lifting Quasianalytic Mappings over Invariants Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 |
5. CJM 2011 (vol 63 pp. 1058)
$S_3$-covers of Schemes We analyze flat $S_3$-covers of schemes, attempting to create
structures parallel to those found in the abelian and triple cover
theories. We use an initial local analysis as a guide in finding a
global description.
Keywords:nonabelian groups, permutation group, group covers, schemes Category:14L30 |
6. CJM 2011 (vol 63 pp. 878)
The Toric Geometry of Triangulated Polygons in Euclidean Spac Speyer and Sturmfels associated GrÃ¶bner toric
degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$
of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each
trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations
induce toric
degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the
space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line.
Our goal in this paper is to give a
geometric (Euclidean polygon) description of the toric fibers
and describe the action of the
compact part of the torus
as "bendings of polygons".
We prove the conjecture of Foth and Hu that
the toric fibers are homeomorphic
to the spaces defined by Kamiyama and Yoshida.
Categories:14L24, 53D20 |
7. CJM 2009 (vol 62 pp. 473)
GoreskyâMacPherson Calculus for the Affine Flag Varieties We use the fixed point arrangement technique developed by
Goresky and MacPherson to calculate the part of the
equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated
by degree 2. We use this result to show that the vertices of the
moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid.
Categories:14L30, 55N91 |
8. CJM 2009 (vol 62 pp. 262)
On the Spectrum of the Equivariant Cohomology Ring If an algebraic torus $T$ acts on a complex projective algebraic
variety $X$, then the affine scheme $\operatorname{Spec}
H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is
often an arrangement of linear subspaces of the vector space
$H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring
of $X$ can be described in terms of this arrangement.
Categories:14L30, 54H15 |
9. CJM 2009 (vol 61 pp. 1407)
Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$ In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.
Categories:14L24, 22E40, 32M15, 51M10 |
10. 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 |
11. CJM 2008 (vol 60 pp. 556)
Polarization of Separating Invariants We prove a characteristic free version of Weyl's theorem on
polarization. Our result is an exact analogue of Weyl's theorem, the
difference being that our statement is about separating invariants
rather than generating invariants. For the special case of finite
group actions we introduce the concept of \emph{cheap polarization},
and show that it is enough to take cheap polarizations of invariants
of just one copy of a representation to obtain separating vector
invariants for any number of copies. This leads to upper bounds on
the number and degrees of separating vector invariants of finite
groups.
Keywords:Jan Draisma, Gregor Kemper, David Wehlau Categories:13A50, 14L24 |
12. CJM 2008 (vol 60 pp. 109)
Affine Lines on Affine Surfaces and the Makar--Limanov Invariant A smooth affine surface $X$ defined over the complex field $\C$ is an $\ML_0$ surface if the
Makar--Limanov invariant $\ML(X)$ is trivial. In this paper we study the topology and geometry of
$\ML_0$ surfaces. Of particular interest is the question: Is every curve $C$ in $X$ which is isomorphic
to
the affine line a fiber component of an $\A^1$-fibration
on $X$? We shall show that the answer is affirmative if the Picard number
$\rho(X)=0$, but negative in case $\rho(X) \ge 1$. We shall also study the ascent and descent of
the $\ML_0$ property under proper maps.
Categories:14R20, 14L30 |
13. CJM 2006 (vol 58 pp. 1000)
On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of $\operatorname{SL}_n$ We compute some Hodge and Betti numbers of the moduli space of stable rank $r$,
degree $d$ vector bundles on a smooth projective curve. We
do not assume $r$ and $d$ are coprime.
In the process we equip the cohomology of an arbitrary algebraic stack with a
functorial mixed Hodge structure. This Hodge structure is
computed in the case of the moduli stack of rank $r$, degree
$d$ vector bundles on a curve. Our methods also yield a formula
for the Poincar\'e
polynomial of the moduli stack that is valid over any
ground field. In the last section we use the previous sections
to give a proof that the Tamagawa number of $\sln$ is one.
Categories:14H, 14L |
14. CJM 2006 (vol 58 pp. 93)
Motivic Haar Measure on Reductive Groups We define a motivic analogue of the Haar measure for groups of the form
$G(k\llp t\rrp)$, where~$k$ is an algebraically closed field
of characteristic zero, and $G$ is a reductive algebraic group defined over
$k$.
A classical Haar measure on such groups does not
exist since they are not locally compact.
We use the theory of motivic integration introduced by M.~Kontsevich to
define an additive function on a certain natural Boolean algebra of subsets of
$G(k\llp t\rrp)$. This function takes values in the so-called dimensional
completion of
the Grothendieck ring of the category of varieties over the base
field. It is invariant under translations by all elements of $G(k\llp t\rrp)$,
and therefore we call it a motivic analogue of Haar measure.
We give an explicit construction of the motivic Haar measure, and then prove
that the result is independent of all the choices that are made in the process.
Keywords:motivic integration, reductive group Categories:14A15, 14L15 |
15. CJM 2003 (vol 55 pp. 693)
Une formule de Riemann-Roch Ã©quivariante pour les courbes Soit $G$ un groupe fini agissant sur une courbe alg\'ebrique
projective et lisse $X$ sur un corps alg\'ebriquement clos $k$. Dans
cet article, on donne une formule de Riemann-Roch pour la
caract\'eristique d'Euler \'equivariante d'un $G$-faisceau inversible
$\mathcal{L}$, \`a valeurs dans l'anneau $R_k (G)$ des caract\`eres du
groupe $G$. La formule donn\'ee a un bon comportement fonctoriel en
ce sens qu'elle rel\`eve la formule classique le long du morphisme
$\dim \colon R_k (G) \to \mathbb{Z}$, et est valable m\^eme pour une
action sauvage. En guise d'application, on montre comment calculer
explicitement le caract\`ere de l'espace des sections globales d'une
large classe de $G$-faisceaux inversibles, en s'attardant sur le cas
particulier d\'elicat du faisceau des diff\'erentielles sur la courbe.
Keywords:group actions on varieties or schemes,, Riemann-Roch theorems Categories:14L30, 14C40 |
16. CJM 2002 (vol 54 pp. 595)
Lie Algebras of Pro-Affine Algebraic Groups We extend the basic theory of Lie algebras of affine algebraic groups
to the case of pro-affine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the pro-discrete
topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine
algebraic group $G$ over $K$, which is discrete in the
finite-dimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
Categories:14L, 16W, 17B45 |
17. CJM 2002 (vol 54 pp. 554)
Equivariant Embeddings into Smooth Toric Varieties We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a well-known embedding theorem
of Sumihiro on quasiprojective $G$-varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$-variety to
a more general framework.
Categories:14E25, 14C20, 14L30, 14M25 |
18. CJM 2000 (vol 52 pp. 1018)
Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$-equivariant centers, into a $G$-variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.
Categories:14L30, 14E15, 14E05, 12E05, 20G10 |
19. CJM 1999 (vol 51 pp. 771)
Stable Bi-Period Summation Formula and Transfer Factors This paper starts by introducing a bi-periodic summation formula
for automorphic forms on a group $G(E)$, with periods by a subgroup
$G(F)$, where $E/F$ is a quadratic extension of number fields. The
split case, where $E = F \oplus F$, is that of the standard trace
formula. Then it introduces a notion of stable bi-conjugacy, and
stabilizes the geometric side of the bi-period summation formula.
Thus weighted sums in the stable bi-conjugacy class are expressed
in terms of stable bi-orbital integrals. These stable integrals
are on the same endoscopic groups $H$ which occur in the case of
standard conjugacy.
The spectral side of the bi-period summation formula involves
periods, namely integrals over the group of $F$-adele points of
$G$, of cusp forms on the group of $E$-adele points on the group
$G$. Our stabilization suggests that such cusp forms---with non
vanishing periods---and the resulting bi-period distributions
associated to ``periodic'' automorphic forms, are related to
analogous bi-period distributions associated to ``periodic''
automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$.
This offers a sharpening of the theory of liftings, where periods
play a key role.
The stabilization depends on the ``fundamental lemma'', which
conjectures that the unit elements of the Hecke algebras on $G$ and
$H$ have matching orbital integrals. Even in stating this
conjecture, one needs to introduce a ``transfer factor''. A
generalization of the standard transfer factor to the bi-periodic
case is introduced. The generalization depends on a new definition
of the factors even in the standard case.
Finally, the fundamental lemma is verified for $\SL(2)$.
Categories:11F72, 11F70, 14G27, 14L35 |
20. CJM 1999 (vol 51 pp. 616)
Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory Let $L$ be a simple algebraic group and $P$ a parabolic subgroup
with Abelian unipotent radical $P^u$. Many familiar varieties
(determinantal varieties, their symmetric and skew-symmetric
analogues) arise as closures of $P$-orbits in $P^u$. We give a
unified invariant-theoretic treatment of various properties of
these orbit closures. We also describe the closures of the
conormal bundles of these orbits as the irreducible components of
some commuting variety and show that the polynomial algebra
$k[P^u]$ is a free module over the algebra of covariants.
Categories:14L30, 13A50 |
21. CJM 1998 (vol 50 pp. 929)
Decomposition varieties in semisimple Lie algebras The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 |
22. CJM 1998 (vol 50 pp. 378)
Equivariant polynomial automorphism of $\Theta$-representations We show that every equivariant polynomial automorphism of a
$\Theta$-repre\-sen\-ta\-tion and of the reduction of an irreducible
$\Theta$-representation is a multiple of the identity.
Categories:14L30, 14L27 |