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Register a new userQuasi-Invariants in Characteristic $p$ and Twisted Quasi-Invariants
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The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.
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Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^* denotes rational singular cohomology, in the case when G is generated by reflections in V and A is the set of reflecting hyperplanes determined by G, or a closely related arrangement. Our main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to leading to proof of the description of the space of invariants conjectured by Felder and Veselov for Coxeter groups that does not rely on computer calculations, this construction provides an extension of the description of the space of invariants proposed by Felder and Veselov to arbitrary finite unitary reflection groups.
We extend the gravitational self-force approach to encompass `self-interaction tidal effects for a compact body of mass $mu$ on a quasi-circular orbit around a black hole of mass $M gg mu$. Specifically, we define and calculate at $O(mu)$ (conservative) shifts in the eigenvalues of the electric- and magnetic-type tidal tensors, and a (dissipative) shift in a scalar product between their eigenbases. This approach yields four gauge-invariant functions, from which one may construct other tidal quantities such as the curvature scalars and the speciality index. First, we analyze the general case of a geodesic in a regular perturbed vacuum spacetime admitting a helical Killing vector and a reflection symmetry. Next, we specialize to focus on circular orbits in the equatorial plane of Kerr spacetime at $O(mu)$. We present accurate numerical results for the Schwarzschild case for orbital radii up to the light-ring, calculated via independent implementations in Lorenz and Regge-Wheeler gauges. We show that our results are consistent with leading-order post-Newtonian expansions, and demonstrate the existence of additional structure in the strong-field regime. We anticipate that our strong-field results will inform (e.g.) effective one-body models for the gravitational two-body problem that are invaluable in the ongoing search for gravitational waves.
Let $K$ be an algebraically closed field of characteristic zero. Algebraic structures of a specific type (e.g. algebras or coalgebras) on a given vector space $W$ over $K$ can be encoded as points in an affine space $U(W)$. This space is equipped with a $text{GL}(W)$ action, and two points define isomorphic structures if and only if they lie in the same orbit. This leads to study the ring of invariants $K[U(W)]^{text{GL}(W)}$. We describe this ring by generators and relations. We then construct combinatorially a commutative ring $K[X]$ which specializes to all rings of invariants of the form $K[U(W)]^{text{GL}(W)}$. We show that the commutative ring $K[X]$ has a richer structure of a Hopf algebra with additional coproduct, grading, and an inner product which makes it into a rational PSH-algebra, generalizing a structure introduced by Zelevinsky. We finish with a detailed study of $K[X]$ in the case of an algebraic structure consisting of a single endomorphism, and show how the rings of invariants $K[U(W)]^{text{GL}(W)}$ can be calculated explicitly from $K[X]$ in this case.
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.
For a finite group $G$, let $p(G)$ denote the minimal degree of a faithful permutation representation of $G$. The minimal degree of a faithful representation of $G$ by quasi-permutation matrices over the fields $mathbb{C}$ and $mathbb{Q}$ are denoted by $c(G)$ and $q(G)$ respectively. In general $c(G)leq q(G)leq p(G)$ and either inequality may be strict. In this paper, we study the representation theory of the group $G =$ Hol$(C_{p^{n}})$, which is the holomorph of a cyclic group of order $p^n$, $p$ a prime. This group is metacyclic when $p$ is odd and metabelian but not metacyclic when $p=2$ and $n geq 3$. We explicitly describe the set of all isomorphism types of irreducible representations of $G$ over the field of complex numbers $mathbb{C}$ as well as the isomorphism types over the field of rational numbers $mathbb{Q}$. We compute the Wedderburn decomposition of the rational group algebra of $G$. Using the descriptions of the irreducible representations of $G$ over $mathbb{C}$ and over $mathbb{Q}$, we show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$. The proofs are often different for the case of $p$ odd and $p=2$.
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