Exterior Algebra Research Articles

Explicit Pieri Inclusions

By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fløystad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups.
 In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.

An Exterior Algebraic Derivation of the Euler–Lagrange Equations from the Principle of Stationary Action

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

Finiteness conditions for the box-tensor product of groups and related constructions

We study finiteness conditions and closure properties for the box-tensor product G⊠H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G⊗H. In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H2(G), the non-abelian exterior square G∧G and the second stable homotopy group of an Eilenberg-MacLane space π2S(K(G,1)).

Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$: simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

Noncommutative matrix factorizations with an application to skew exterior algebras

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.

DGAs with polynomial homology

In this work, we study the classification of differential graded algebras over Z (DGAs) whose homology is Fp[x], i.e. the polynomial algebra over Fp on a single generator. This classification problem was left open in work of Dwyer, Greenlees and Iyengar.For |y2p−2|=2p−2, we show that there is a unique non-formal DGA with homology Fp[y2p−2] and a non-formal 2p−2 Postnikov section. Among a classification result, this provides the first example of a non-formal DGA with homology Fp[x]. By duality, this also shows that there is a non-formal DGA whose homology is an exterior algebra over Fp with a generator in degree −(2p−1).Considering the classification of the ring spectra corresponding to these DGAs, we show that every E2 DGA with homology Fp[x] (with no restrictions on |x|) is topologically equivalent to the formal DGA with homology Fp[x], i.e. they are topologically formal. This follows by a theorem of Hopkins and Mahowald.

A tensor product approach to compute 2-nilpotent multiplier of p-groups

Let [Formula: see text] be a group given by a free presentation [Formula: see text]. The 2-nilpotent multiplier of [Formula: see text] is the abelian group [Formula: see text] which is invariant of [Formula: see text] [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc. 58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of [Formula: see text], when [Formula: see text] is a finite (generalized) extra special [Formula: see text]-group. Moreover, the descriptions of the triple tensor product [Formula: see text], and the triple exterior product [Formula: see text] are given.

Approximate Counting of k -Paths: Simpler, Deterministic, and in Polynomial Space

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).

Combinatorics in the exterior algebra and the Bollobás Two Families Theorem

We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $\tau$ from $A$ to $B$, and $\tau$ is strictly better than another house allocation $\tau'\neq \tau$ if for every buyer $i$, $\tau'(i)$ does not come before $\tau(i)$ in the preference list of $i$. A house allocation is Pareto optimal if there is no strictly better house allocation. Let $s(\tau)$ be the image of $\tau$ (i.e., the set of houses sold in the house allocation $\tau$). We are interested in the largest possible cardinality $f(m)$ of the family of sets $s(\tau)$ for Pareto optimal mappings $\tau$ taken over all sets of preference lists of $m$ buyers. We improve the earlier upper bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection between this problem and some problems in extremal set theory.

Fukaya’s conjecture on Witten’s twisted $A_\infty$ structure

The wedge product on de Rham complex of a Riemannian manifold $M$ can be pulled back to $H^\ast (M)$ via explicit homotopy constructed by using Green’s operator which gives higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures has semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of de Rham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya in [6].

Interior product, Lie derivative and Wilson line in the KBc subsector of open string field theory

The open string field theory of Witten (SFT) has a close formal similarity with Chern-Simons theory in three dimensions. This similarity is due to the fact that the former theory has concepts corresponding to forms, exterior derivative, wedge product and integration over the manifold. In this paper, we introduce the interior product and the Lie derivative in the KBc subsector of SFT. The interior product in SFT is specified by a two-component “tangent vector” and lowers the ghost number by one (like the ordinary interior product maps a p-form to (p − 1)-form). The Lie derivative in SFT is defined as the anti-commutator of the interior product and the BRST operator. The important property of these two operations is that they respect the KBc algebra.Deforming the original (K, B, c) by using the Lie derivative, we can consider an infinite copies of the KBc algebra, which we call the KBc manifold. As an application, we construct the Wilson line on the manifold, which could play a role in reproducing degenerate fluctuation modes around a multi-brane solution.

Lelong–Demailly Numbers and Wedge Product of S-Plurisubahrmonic Currents

Lelong–Demailly Numbers and Wedge Product of S-Plurisubahrmonic Currents

The non-abelian tensor and exterior products of crossed modules of Lie superalgebras

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras.

Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping

In this paper, the symplectic perturbation series methodology of the non-conservative linear Hamiltonian system is presented for the structural dynamic response with damping. Firstly, the linear Hamiltonian system is briefly introduced and its conservation law is proved based on the properties of the exterior products. Then the symplectic perturbation series methodology is proposed to deal with the non-conservative linear Hamiltonian system and its conservation law is further proved. The structural dynamic response problem with eternal load and damping is transformed as the non-conservative linear Hamiltonian system and the symplectic difference schemes for the non-conservative linear Hamiltonian system are established. The applicability and validity of the proposed method are demonstrated by three engineering examples. The results demonstrate that the presented methodology is better than the traditional Runge–Kutta algorithm in the prediction of long-time structural dynamic response under the same time step.

A simple and complete discrete exterior calculus on general polygonal meshes

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

Divergence-free quantum electrodynamics in locally conformally flat space–time

This paper describes an approach to quantum electrodynamics (QED) in curved space–time obtained by considering infinite-dimensional algebra bundles associated to a natural principal bundle [Formula: see text] associated with any locally conformally flat space–time, with typical fibers including the Fock space and a space of fermionic multiparticle states which forms a Grassmann algebra. Both these algebras are direct sums of generalized Hilbert spaces. The requirement of [Formula: see text] covariance associated with the geometry of space–time, where [Formula: see text] is the structure group of [Formula: see text], leads to the consideration of [Formula: see text] intertwining operators between various spaces. Scattering processes are associated with such operators and are encoded in an algebra of kernels. Intertwining kernels can be generated using [Formula: see text] covariant matrix-valued measures. Feynman propagators, fermion loops and the electron self-energy can be given well-defined interpretations as such measures. Divergence-free calculations in QED can be carried out by computing the spectra of these measures and kernels (a process called spectral regularization). As an example of the approach the precise Uehling potential function for the [Formula: see text] atom is calculated without requiring renormalization from which the Uehling contribution to the Lamb shift can be calculated exactly.

Cohomological supports over derived complete intersections and local rings

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples; one of which answers a question of D. Jorgensen in the negative.

The reverse decomposition of unipotents for bivectors

For the second fundamental representation of the general linear group over a commutative ring R, we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse. Toward the solution, we get stabilization theorems for any column of a matrix from or from the exterior square of

Capability of Nilpotent Lie Algebras of Small Dimension

Given a nilpotent Lie algebra $L$ of dimension $\le 6$ on an arbitrary field of characteristic $\neq 2$, we show a direct method which allows us to detect the capability of $L$ via computations on the size of its nonabelian exterior square $L \wedge L$. For dimensions higher than $ 6$, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of $L \wedge L$ via the size of the Schur multiplier $M(L/Z^\wedge(L))$ of $L/Z^\wedge(L)$, where $Z^\wedge(L)$ denotes the exterior center of $L$.

Decomposition of exterior and symmetric squares in characteristic two

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e∈sl(V), we describe the Jordan normal form of e on the sl(V)-modules ∧2(V) and S2(V). In the case where e is a regular nilpotent element, we are able to give a closed formula.We also consider the closely related problem of describing, for every unipotent element u∈SL(V), the Jordan normal form of u on ∧2(V) and S2(V). A recursive formula for the Jordan block sizes of u on ∧2(V) was given by Gow and Laffey [J. Group Theory 9 (2006), 659–672]. We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S2(V).