Related papers: Crystal invariant theory I: Geometric RSK
Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend…
We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction…
In this article we study the structure of $\Gamma$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $\Gamma$, a non commutative group in the form of a semidirect…
From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra.…
In this paper we construct an algebraic invariant attached to Galois representations over number fields. This invariant, which we call an Artin symmetric function, lives in a certain ring we introduce called the ring of arithmetic symmetric…
The fingerprint invariant of partitions can be used to describe the Kazhdan-Lusztig map for the classical groups. We discuss the basic properties of fingerprint. We construct the fingerprints of rigid partitions in the $B_n$, $C_n$, and…
One of basic difficulties of machine learning is handling unknown rotations of objects, for example in image recognition. A related problem is evaluation of similarity of shapes, for example of two chemical molecules, for which direct…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{sl}_2)$ as a cluster transformation and introducing an auxiliary parameter $\epsilon$, we derive a…
We construct quantum K-invariants in non-archimedean analytic geometry. Contrary to the classical approach in algebraic geometry via perfect obstruction theory, we build on our previous works on the foundations of derived non-archimedean…
Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with…
Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was…
In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction…
V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula…
The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic $p$ with the one-dimensional sign representation. It can also be interpreted as an…
It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However their relation to the symmetry group of diffeomorphism…
Already Hermann Grassmann's father Justus (1829, 1830) published two works on the geometrical description of crystals, influenced by the earlier works of C.S. Weiss (1780-1856) on three main crystal forces governing crystal formation. In…
The symbol invariant is used to describe the Springer correspondence for the classical groups by Lusztig. And the fingerprint invariant can be used to describe the Kazhdan-Lusztig map. They are invariants of rigid semisimple operators…