Quotients, automorphisms and differential operators

Let $V$ be a $G$-module where $G$ is a complex reductive group. Let Z:=\quot VG denote the categorical quotient and let $\pi\colon V\to Z$ be the morphism dual to the inclusion \O(V)^G\subset\O(V). Let $\phi\colon Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $\Phi\colon V\to V$ which lifts $\phi$, i.e., $\pi(\Phi(v))=\phi(\pi(v))$ for all $v\in V$. In \cite{Kuttler} the case is treated where V=r\lieg is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\geq 2$ will do), any $\phi$ has a lift. We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of $Z$, and we show that if a holomorphic $\phi$ and its inverse lift holomorphically, then $\phi$ has a lift $\Phi$ which is an automorphism such that $\Phi(gv)=\sigma(g)\Phi(v)$, $v\in V$, $g\in G$ where $\sigma$ is an automorphism of $G$. We reduce the lifting problem to the group of automorphisms of $Z$ which preserve the natural grading of \O(Z)\simeq\O(V)^G. Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler's methods to show lifting in case $V$ contains a copy of \lieg.Comment: 23 pages, minor revisions. To appear in J. London Math. Societ

Quotient p-Schatten metrics on spheres

Let S(H) be the unit sphere of a Hilbert space H and Up(H) thegroup of unitary operators in H such that u−1 belongs to the p-Schatten idealBp(H). This group acts smoothly and transitively in S(H) and endows it witha natural Finsler metric induced by the p-norm kzkp = tr(zz∗)p/21/p. Thismetric is given bykvkx,p = min{kz − ykp : y ∈ gx},where z ∈ Bp(H)ah satisfies that (dπx)1(z) = z · x = v and gx denotes theLie algebra of the subgroup of unitaries which fix x. We call z a lifting of v.A lifting z0 is called a minimal lifting if additionally kvkx,p = kz0kp. Inthis paper we show properties of minimal liftings and we treat the problemof finding short curves α such that α(0) = x and ˙α(0) = v with x ∈ S(H)and v ∈ TxS(H) given. Also we consider the problem of finding short curveswhich join two given endpoints x, y ∈ S(H).Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Antunez, Andrea. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentin

Navigating in the Cayley graph of SL2(Fp)SL_2(F_p) and applications to hashing

Cayley hash functions are based on a simple idea of using a pair of (semi)group elements, $A$ and $B$, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the (semi)group. In this paper, we focus on hashing with $2 \times 2$ matrices over $F_p$. Since there are many known pairs of $2 \times 2$ matrices over $Z$ that generate a free monoid, this yields numerous pairs of matrices over $F_p$, for a sufficiently large prime $p$, that are candidates for collision-resistant hashing. However, this trick can "backfire", and lifting matrix entries to $Z$ may facilitate finding a collision. This "lifting attack" was successfully used by Tillich and Z\'emor in the special case where two matrices $A$ and $B$ generate (as a monoid) the whole monoid $SL_2(Z_+)$. However, in this paper we show that the situation with other, "similar", pairs of matrices from $SL_2(Z)$ is different, and the "lifting attack" can (in some cases) produce collisions in the group generated by $A$ and $B$, but not in the positive monoid. Therefore, we argue that for these pairs of matrices, there are no known attacks at this time that would affect security of the corresponding hash functions. We also give explicit lower bounds on the length of collisions for hash functions corresponding to some particular pairs of matrices from $SL_2(F_p)$.Comment: 10 page

Arithmetical Congruence Preservation: from Finite to Infinite

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $\pi\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $\pi\circ F=f\circ\pi$. In this paper we relate the finite and infinite notions by proving that the finite case can be lifted to the infinite one. For $p$-adic and profinite integers we get similar characterizations via lifting. We also prove that lattices of recognizable subsets of $Z$ are stable under inverse image by congruence preserving functions

State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions

For the strictly positive case (the suboptimal case), given stable rational matrix functions $G$ and $K$, the set of all $H^\infty$ solutions $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions $G$ and $K$. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.Comment: 28 page