Exterior Algebra Research Articles

Erdős–Ko–Rado and Hilton–Milner theorems for two-forms

The celebrated Erdős–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of {1,…,n} was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erdős–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines.

Basic 𝕋-Spaces in the Relatively Free Grassmann Algebra Without Unity

In this paper, we consider the 𝕋-space structure of the relatively free Grassmann algebra 𝔽 (3) without unity over an infinite field of prime and zero characteristic. Our work is focused on 𝕋-spaces 𝕎n generated by all so-called n-words. A question about connections between 𝕎r and 𝕎n for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions, which, to some extent, clarify the structure of the algebra: the basic 𝕋-spaces produce infinite strictly descending chains of inclusions in the algebra 𝔽(3).

The exterior graded Swiss-Cheese Operad

ABSTRACT In this paper we introduce a generalization of the exterior algebra of a vector space V. We study some of its properties and give explicit computations for the case . In particular we show the existence and uniqueness of a determinant-like function with the property that if there exists such that . We also introduce the notion of GSC-operads and show how fits in that context.

A POISSON ALGEBRA STRUCTURE OVER THE EXTERIOR ALGEBRA OF A QUADRATIC SPACE

We construct a Poisson algebra structure of degree $-2$ over the exterior algebra of a quadratic space. Here we do not use Clifford algebra as in [4].

New incompressible symmetric tensor categories in positive characteristic

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If char(k)=p>0, then we use this method to construct the incompressible abelian symmetric tensor categories Verpn, Verpn+ generalizing earlier constructions by Gelfand–Kazhdan and Georgiev–Mathieu for n=1, and by Benson–Etingof for p=2. Namely, Verpn is the abelian envelope of the quotient of the category of tilting modules for SL2(k) by the nth Steinberg module, and Verpn+ is its subcategory generated by PGL2(k)-modules. We show that Verpn are reductions to characteristic p of Verlinde braided tensor categories in characteristic 0, which explains the notation. We study the structure of these categories in detail and, in particular, show that they categorify the real cyclotomic rings Z[2cos(2π∕pn)], and that Verpn embeds into Verpn+1. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Verp∞ of the nested sequence Verp⊂Verp2⊂⋯ . This would provide an analogue of Deligne’s theorem in characteristic 0 and a generalization of the results of Coulembier, Etingof, and Ostrik, which shows that this conjecture holds for Frobenius exact (in particular, semisimple) categories, and, moreover, the fiber functor lands in Verp (in the case of fusion categories, this was shown earlier by Ostrik). Finally, we classify symmetric tensor categories generated by an object with invertible exterior square; this class contains the categories Verpn.

Local Langlands correspondence for the twisted exterior and symmetric square $$\epsilon $$-factors of $$\text {GL}_n$$

Local Langlands correspondence for the twisted exterior and symmetric square $$\epsilon $$-factors of $$\text {GL}_n$$

Quasisymmetric harmonics of the exterior algebra

AbstractWe study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1)The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$ .(2)There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$ .(3)There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.

Automorphisms and superalgebra structures on the Grassmann algebra

Let F be a field of characteristic zero and let E be the Grassmann algebra of an infinite dimensional F-vector space L. In this paper we study the superalgebra structures (that is the Z2-gradings) that the algebra E admits. By using the duality between superalgebras and automorphisms of order at most 2 we prove that in many cases the Z2-graded polynomial identities for such structures coincide with the Z2-graded polynomial identities of the “typical” cases E∞, Ek⁎ and Ek where the vector space L is homogeneous. Recall that these cases were completely described by Di Vincenzo and da Silva in [13]. Moreover we exhibit a wide range of non-homogeneous Z2-gradings on E that are Z2-isomorphic to E∞, Ek⁎ and Ek. In particular we construct a Z2-grading on E with only one homogeneous generator in L which is Z2-isomorphic to the natural Z2-grading on E, here denoted by Ecan.

Basic 𝕋-Spaces in the Relatively Free Grassmann Algebra Without Unity

In this paper, we consider the 𝕋-space structure of the relatively free Grassmann algebra 𝔽(3) without unity over an infinite field of prime and zero characteristic. Our work is focused on 𝕋-spaces 𝕎n generated by all n-words. A question about connections between 𝕎r and 𝕎n for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions that, to some extent, clarify the structure of the algebra: the basic 𝕋-spaces produce infinite strictly descending chains of inclusions in the algebra 𝔽(3).

Overgroups of exterior powers of an elementary group. levels

ABSTRACT We prove a first part of the standard description of groups H lying between an exterior power of an elementary group m E n ⁡ ( R ) and a general linear group GL ( n m ) ⁡ ( R ) for a commutative ring R , 2 ∈ R ∗ and n ⩾ 3 m . The description uses the classical notion of a level: for every group H we find a unique ideal A of the ground ring R, which describes H.

A Generalized Newton Method for Subgradient Systems

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second order subdifferential of such functions that enjoy extensive calculus rules and can be efficiently computed for broad classes of extended real-valued functions. Based on this and on the metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring the well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian gradients ([Formula: see text] functions), which is the underlying case of our consideration. The developed algorithms for prox-regular functions and their extension to a structured class of composite functions are formulated in terms of proximal mappings and forward–backward envelopes. Besides numerous illustrative examples and comparison with known algorithms for [Formula: see text] functions and generalized equations, the paper presents applications of the proposed algorithms to regularized least square problems arising in statistics, machine learning, and related disciplines. Funding: Research of P. D. Khanh is funded by Ho Chi Minh City University of Education Foundation for Science and Technology [Grant CS.2022.19.20TD]. Research of B. Mordukhovich and V. T. Phat was partly supported by the U.S. National Science Foundation [Grants DMS-1808978 and DMS-2204519]. The research of B. Mordukhovich was also supported by the Air Force Office of Scientific Research [Grant 15RT0462] and the Australian Research Council under Discovery Project DP-190100555. This work was supported by the Air Force Office of Scientific Research [Grant 15RT0462].

Transitive irreducible Lie superalgebras of vector fields

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.

The fermionic integral on loop space and the Pfaffian line bundle

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the “top degree component” of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible “top degree component” to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space. This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space.

A quantum shuffle approach to quantum determinants

Let $\bigwedge_\sigma V=\bigoplus_{k\geq 0}\bigwedge_\sigma^kV$ be the quantum exterior algebra associated to a finite-dimensional braided vector space $(V,\sigma)$. For an algebra $\mathfrak{A}$, we consider the convolution product on the graded space $\bigoplus_{k\geq 0}\mathrm{Hom}\big(\bigwedge_\sigma^kV,\bigwedge_\sigma^kV\otimes \mathfrak{A}\big)$. Using this product, we define a notion of quantum minor determinant of a map from $V$ to $V\otimes \mathfrak{A}$, which coincides with the classical one in the case that $\mathfrak{A}$ is the FRT algebra corresponding to $U_q(\mathfrak{sl}_N)$. We establish quantum Laplace expansion formulas and multiplicative formulas for these determinants.

Coxeter factorizations with generalized Jucys–Murphy weights and Matrix‐Tree theorems for reflection groups

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group W $W$ , in terms of the spectrum of an associated operator, the W $W$ -Laplacian. This covers in particular all finite Coxeter groups. The results of this paper include generalizations of the Matrix-Tree and Matrix Forest theorems to reflection groups, and cover reduced (shortest length) as well as arbitrary length factorizations. Our formulas are relative to a choice of weighting system that consists of n $n$ free scalar parameters and is defined in terms of a tower of parabolic subgroups. To study such systems we introduce (a class of) variants of the Jucys–Murphy elements for every group, from which we define a new notion of “tower equivalence” of virtual characters. A main technical point is to prove the tower equivalence between virtual characters naturally appearing in the problem, and exterior products of the reflection representation of W $W$ . Finally we study how this W $W$ -Laplacian matrix we introduce can be used in other problems in Coxeter combinatorics; for instance, we explain how it defines analogs of trees for W $W$ and how it relates them to Coxeter factorizations. Moreover, we build a parabolic recursion for the W $W$ -Laplacian that proves, without relying on the classification, new numerological identities between the Coxeter number of W $W$ and those of its parabolic subgroups, and, when W $W$ is a Weyl group, a new, explicit formula for the volume of the corresponding root zonotope.

Resolutions of ideals of subspace arrangements

Given a collection of t subspaces in an n-dimensional vector space W we can associate to them t linear ideals in the symmetric algebra 𝒮(W∗). Conca and Herzog showed that the Castelnuovo–Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo–Mumford regularity of the intersection of t linear ideals is at most t. We show that analogous results hold when we work over the exterior algebra ⋀(W∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.

Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.

Exterior algebras in matroid theory

Ordered blueprints are algebraic objects that generalize monoids and ordered semirings, and mathbb {F}_1^{pm }-algebras are ordered blueprints that have an element epsilon that acts as -1. In this work we introduce an analogue of the exterior algebra for mathbb {F}_1^{pm }-algebras that provides a new cryptomorphism for matroids. We also show how to recover the usual exterior algebra if the mathbb {F}_1^{pm }-algebra comes from a ring, and the Giansiracusa Grassmann algebra if the mathbb {F}_1^{pm }-algebra comes from an idempotent semifield.

Some achieved wedge products of positive currents

In this paper, we study the existence of the current gT for positive plurisubharmonic currents T and unbounded plurisubharmonic functions g.

On the L∞ structure of Poisson gauge theory

The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory. We derive the minimal set of non-vanishing ℓ-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L∞ algebras. On the other hand, it can be used as a starting point for bootstrapping the full non-commutative gauge theory. The first few brackets of such a theory are constructed explicitly in the text. In addition we show that the derivation properties of ℓ-brackets on with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, does not have a structure of P∞ algebra.