Algebraic Invariants Research Articles

Computational methods for t-spread monomial ideals

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ a standard polynomial ring over $K$. In this paper, we give new combinatorial algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a given set of $t$-spread monomials of $S$. Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a \emph{Macaulay2} package, \texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.

Ultrafast learning of four-node hybridization cycles in phylogenetic networks using algebraic invariants.

The abundance of gene flow in the Tree of Life challenges the notion that evolution can be represented with a fully bifurcating process which cannot capture important biological realities like hybridization, introgression, or horizontal gene transfer. Coalescent-based network methods are increasingly popular, yet not scalable for big data, because they need to perform a heuristic search in the space of networks as well as numerical optimization that can be NP-hard. Here, we introduce a novel method to reconstruct phylogenetic networks based on algebraic invariants. While there is a long tradition of using algebraic invariants in phylogenetics, our work is the first to define phylogenetic invariants on concordance factors (frequencies of four-taxon splits in the input gene trees) to identify level-1 phylogenetic networks under the multispecies coalescent model. Our novel hybrid detection methodology is optimization-free as it only requires the evaluation of polynomial equations, and as such, it bypasses the traversal of network space, yielding a computational speed at least 10 times faster than the fastest-to-date network methods. We illustrate our method's performance on simulated and real data from the genus Canis. We present an open-source publicly available Julia package PhyloDiamond.jl available at https://github.com/solislemuslab/PhyloDiamond.jl with broad applicability within the evolutionary community.

Spectral invariance of quasi-Banach algebras of matrices and pseudodifferential operators

Abstract We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 {p<1} . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand’s results.

Extended Brauer analysis of some Dynkin and Euclidean diagrams

<p>The analysis of algebraic invariants of algebras induced by appropriated multiset systems called Brauer configurations is a Brauer analysis of the data defining the multisets. Giving a complete description of such algebraic invariants (e.g., giving a closed formula for the dimensions of algebras induced by significant classes of Brauer configurations) is generally a tricky problem. Ringel previously proposed an analysis of this type in the case of Dynkin algebras, for which so-called Dynkin functions were used to study the numerical behavior of invariants associated with such algebras. This paper introduces two additional tools (the entropy and the covering graph of a Brauer configuration) for Brauer analysis, which is applied to Dynkin and Euclidean diagrams to define Dynkin functions associated with Brauer configuration algebras. Properties of graph entropies defined by the corresponding covering graphs are given to establish relationships between the theory of Dynkin functions, the Brauer configuration algebras theory, and the topological content information theory.</p>

Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

Abstract We develop techniques to construct isomorphisms between simple affine W-algebras and affine vertex algebras at admissible levels. We then apply these techniques to obtain many new, and conjecturally all, admissible collapsing levels for affine W-algebras. In short, if a simple affine W-algebra at a given level is equal to its affine vertex subalgebra generated by the centraliser of an ${\mathfrak {sl}}_2$ -triple associated with the underlying nilpotent orbit, then that level is said to be collapsing. Collapsing levels are important both in representation theory and in theoretical physics. Our approach relies on two fundamental invariants of vertex algebras. The first one is the associated variety, which, in the context of admissible level simple affine W-algebras, leads to the Poisson varieties known as nilpotent Slodowy slices. We exploit the singularities of these varieties to detect possible collapsing levels. The second invariant is the asymptotic datum. We prove a general result asserting that, under appropriate hypotheses, equality of asymptotic data implies isomorphism at the level of vertex algebras. Then we use this to give a sufficient criterion, of combinatorial nature, for an admissible level to be collapsing. Our methods also allow us to study isomorphisms between quotients of W-algebras and extensions of simple affine vertex algebras at admissible levels. Based on such examples, we are led to formulate a general conjecture: for any finite extension of vertex algebras, the induced morphism between associated Poisson varieties is dominant.

Differential Invariants of Coupled Hirota-Satsuma KdV Equations

In this paper, we consider a generalized coupled Hirota-Satsuma KdV (CHSK) system of equations. We apply the moving frames method to find a finite generating set of differential invariants for the Lie symmetry group of CHSK equations. Once the generating set of differential invariants is located, we obtain recurrence relations and syzygies among the generating differential invariants. Our approach provides a complete characterization of the structure of algebras of differential invariants of CHSK equations.

Algebraic invariants of edge ideals of some circulant graphs

<abstract><p>Let $ S $ be a polynomial ring over a field and $ I $ be an edge ideal associated with some classes of circulant graphs. We discussed the algebraic invariants, namely, regularity, projective dimension, depth, and the Stanley depth of $ S/I. $</p></abstract>

Realizations and invariants of Lie algebras

Lie algebras of first-order differential operators with smooth coefficients are considered and their construction methods are reviewed. Obstacles preventing contractions of such operators are emphasized and a constructive procedure for parameterizations is proposed. A number of illustrative examples are considered.

On abelian subcategories of triangulated categories

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial t-structures, and in particular, full abelian subcategories need not arise as hearts of a t-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.

Symmetric Polynomials in Free Associative Algebras—II

Let K⟨Xd⟩ be the free associative algebra of rank d≥2 over a field, K. In 1936, Wolf proved that the algebra of symmetric polynomials K⟨Xd⟩Sym(d) is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of K⟨Xd⟩ with the additional action of Sym(n) by permuting the positions of the variables. He proved finite generation with respect to this additional action for the algebra of invariants K⟨Xd⟩G of every reductive group, G. In the first part of the present paper, we established that, over a field of characteristic 0 or of characteristic p>d, the algebra K⟨Xd⟩Sym(d) with the action of Koryukin is generated by (noncommutative version of) the elementary symmetric polynomials. Now we prove that if the field, K, is of positive characteristic at most d then the algebra K⟨Xd⟩Sym(d), taking into account that Koryukin’s action is infinitely generated, describe a minimal generating set.

Syzygies for the vector invariants of the dihedral group

The dihedral group D2n of order 2n acts on the space of m-tuples of vectors from its defining 2-dimensional representation. The corresponding algebra of polynomial invariants has a natural structure as a module over the general linear group GLm(C). Therefore the ideal of relations between the generators of the algebra of invariants can be treated as a GL-ideal (i.e. an ideal stable with respect to the appropriate action of GLm(C)). It is shown that this GL-ideal is generated by relations depending on no more than 3 vector variables. A minimal GL-ideal generating system is found for the cases of m=2 and an arbitrary n, and for the cases of an arbitrary m and n=4, n=5, and n=6.

Hadamard products and binomial ideals

We study the Hadamard product of two varieties V and W, with particular attention to the situation when one or both of V and W is a binomial variety. The main result of this paper shows that when V and W are both binomial varieties, and the binomials that define V and W have the same binomial exponents, then the defining equations of V⋆W can be computed explicitly and directly from the defining equations of V and W. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal IG of a graph G and the toric ideal IH of a subgraph H of G. We also derive results about algebraic invariants of Hadamard products: assuming V and W are binomial with the same exponents, we show that deg⁡(V⋆W)=deg⁡(V)=deg⁡(W) and dim⁡(V⋆W)=dim⁡(V)=dim⁡(W). Finally, given any (not necessarily binomial) projective variety V and a point p∈Pn∖V(x0x1⋯xn), subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points q that satisfy p⋆V=q⋆V.

Symmetry analysis, optimal system, and invariant solutions for a (2+1)-dimensional two-phase mass flow model

The paper investigates a two-phase mass flow model governed by gravity, involving solid particles and a viscous fluid. By utilizing the Lie symmetries admitted by the system, similarity solutions for the (2+1)-dimensional two-phase mass flow model are obtained. A comprehensive set of local point symmetries is established, and a well-suited collection of two-dimensional subalgebras is constructed from the maximal Lie invariance algebra. The optimal system’s vector fields are then utilized to directly reduce the governing model to a system of ordinary differential equations. Through analytical solutions, we successfully solve the resulting systems and further analyze their physical behaviors numerically.

Deformations of Pairs of Kleinian Singularities

Abstract Kleinian singularities, that is, the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory, and Singularity theory. The filtered deformations of these algebras of invariants were classified by Brieskorn (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when the smaller subgroup is normal in the larger.

On Strongest Algebraic Program Invariants

A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location. Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this article, we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result, our main tool is an algebraic result of independent interest: Given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank 1 Lie groups

We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group O 2 ( R ) . The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series. Communicated by Ellen Kirkman

Local Statistics of Random Permutations from Free Products

Abstract Let $\alpha $ and $\beta $ be uniformly random permutations of orders $2$ and $3$, respectively, in $S_{N}$, and consider, say, the permutation $\alpha \beta \alpha \beta ^{-1}$. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let $\Gamma =G_{1}*\ldots *G_{k}$ be a free product of groups where each of $G_{1},\ldots ,G_{k}$ is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element $\gamma \in \Gamma $, a $\gamma $-random permutation in the symmetric group $S_{N}$ is the image of $\gamma $ through a uniformly random homomorphism $\Gamma \to S_{N}$. In this paper we study local statistics of $\gamma $-random permutations and their asymptotics as $N$ grows. We first consider $\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big )\big ]$, the expected number of fixed points in a $\gamma $-random permutation in $S_{N}$. We show that unless $\gamma $ has finite order, the limit of $\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big )\big ]$ as $N\to \infty $ is an integer, and is equal to the number of subgroups $H\le \Gamma $ containing $\gamma $ such that $H\cong \mathbb{Z}$ or $H\cong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $H\le \Gamma $ containing $\gamma $ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big )\big ]$ and determine the limit distribution of the number of fixed points as $N\to \infty $. These results are then generalized to all statistics of cycles of fixed lengths.

The moduli space of G-algebras

Let L be a Galois algebra with Galois group G and let x be a normal element of L. The moduli space X of pairs (L,x) is isomorphic to an open subset of the quotient variety P/G, where P is the projective space of the regular representation of G. We provide a formula for the height of any pair (L,x)∈X(Q) in terms of algebraic invariants of L and x with respect to a natural adelic metric on the anticanonical divisor of P/G.

A new look at Lie algebras

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.