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A formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions, in accordance with definition provided in exterior algebra. [ABSTRACT FROM AUTHOR]

In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra structures. The Clifford-Weyl superalgebras can be realized as tensor product of the algebra of alternating and symmetric tensors respectively, on the even and odd parts of their underlying superspace. For Physical applications in elementaryparticles, we add star structures to these algebras and investigate the basic relations. Ortho-symplectic Lie algebras are naturally present in these algebras and their representations on these algebras can be described easily.

We study the category of GL -equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem à la Nagpal. Using this, we obtain a Church–Ellenberg type bound for the Castelnuovo–Mumford regularity. We also prove finiteness results for local cohomology. [ABSTRACT FROM AUTHOR]

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals. [ABSTRACT FROM AUTHOR]

The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric rank-2 tensors), and they are related by a rank-4 inertia tensor. Remarkably, this inertia tensor belongs to a well-studied class of algebraic curvature tensors that have the same index symmetries as the Riemann curvature tensor field used in general relativity. Any algebraic curvature tensor can be decomposed into irreducible representations of the orthogonal group via the Ricci decomposition. We calculate the Ricci decomposition of the inertia tensor for a rigid body in any number of dimensions, and we find that (unlike for the Riemann curvature tensor field) its traceless Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the traceless Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in n ≥ 4 dimensions. [ABSTRACT FROM AUTHOR]

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weaklyH-saturated in F if the edges in E (F) \ E (G) admit an ordering e 1 , ... , e k so that for all i ∈ [ k ] the hypergraph G ∪ { e 1 , ... , e i } contains an isomorphic copy of H which in turn contains the edge e i . The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F. Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al. [ABSTRACT FROM AUTHOR]

In this paper, we determine the Gerstenhaber Lie bracket and the Batalin–Vilkovisky structure over the Hochschild cohomology ring of exterior algebras. [ABSTRACT FROM AUTHOR]

Given an exterior algebra E over a finite dimension vector space v, and let R = E/I, where I is a graded ideal in E. The relation between the algebra R and Rmon regarding to G-quadratic and LG-quadratic will be investigated. We show that the algebra R is G-quadratic if and only if Rmon is G-quadratic. Furthermore, it has been shown that the algebra R is LG-quadratic if and only if Rmon is LG-quadratic. [ABSTRACT FROM AUTHOR]

In this paper we introduce Λ V S 2 a generalization of the exterior algebra of a vector space V. We study some of its properties and give explicit computations for the case dim (V 2) = 2 . In particular we show the existence and uniqueness of a determinant-like function d e t S 2 : V 2 ⊗ 6 → k with the property that d e t S 2 (⊗ 1 ≤ i < j ≤ 4 (v i , j)) = 0 if there exists 1 ≤ x < y < z ≤ 4 such that v x , y = v x , z = v y , z . We also introduce the notion of GSC-operads and show how Λ V S 2 fits in that context. [ABSTRACT FROM AUTHOR]

In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type D and for the exceptional cases. [ABSTRACT FROM AUTHOR]

For each quadratic form Q ∈ Quad (V) on a vector space over a field K , we can define the Clifford algebra Cl (V , Q) as the quotient T (V) / I (Q) of the tensor algebra T (V) by the two-sided ideal generated by expressions of the form x ⊗ x - Q (x) , x ∈ V. In the present paper we consider the whole family { Cl (V , Q) : Q ∈ Quad (V) } in a geometric way as a Z 2 -graded vector bundle over the base manifold Quad (V). Bilinear forms F ∈ Bil (V) act on this bundle providing natural bijective linear mappings λ ¯ F between different Clifford algebras Cl (V , Q). Alternating (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as 'gauge transformations'. We develop here the formalism of Bourbaki, which generalizes the well known Chevalley's isomorphism Cl (V , Q) → End (⋀ (V)) → ⋀ (V). In particular we realize the Clifford algebra twisting gauge transformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of ⋀ 2 (V ∗) representing these forms. Throughout all this paper we intentionally avoid using the so far accepted term "Clifford algebra of a bilinear form" (known otherwise as "Quantum Clifford algebra"), which we consider as possibly misleading, as it does not represent any well defined mathematical object. Instead we show explicitly how any given Clifford algebra Cl (Q) can be naturally realized as acting via endomorphisms of any other Clifford algebra Cl (Q ′) if Q ′ = Q + Q F , F ∈ Bil (V) and Q F (x) = F (x , x). Possible physical meaning of such transformations is also mentioned. [ABSTRACT FROM AUTHOR]

In this short note, we extend to the continuous case a mean projection theorem for discrete determinantal point processes associated with a finite range projection, thus strengthening a known result in random linear algebra due to Ermakov and Zolotukhin. We also give a new formula for the variance of the exterior power of the random projection. [ABSTRACT FROM AUTHOR]

We give a proof of the Erdős-Ko-Rado Theorem using the Borel Fixed-Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erdős-Ko-Rado Theorem and (generalizations of) the Gerstenhaber Theorem on spaces of nilpotent matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a determinant-like map det S 3 and study some of its properties. For this, we define a graded vector space Λ V S 3 that has similar properties with the exterior algebra Λ V and the exterior GSC-operad Λ V S 2 from Staic. When dim (V 2) = 2 , we show that dim k (Λ V 2 S 3 [ 6 ]) = 1 , which gives the existence and uniqueness of the map det S 3 . We also give an explicit formula for det S 3 as a sum over certain 2-partitions of the complete hypergraph K 6 3 . [ABSTRACT FROM AUTHOR]

The purpose of this short note is to consider multivariate Hasse-Schmidt derivations on exterior algebras and to show how they easily provide remarkable identities, holding in the algebra of square matrices, which generalize the classical theorem of Cayley–Hamilton. [ABSTRACT FROM AUTHOR]

This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative. [ABSTRACT FROM AUTHOR]

In this paper we prove the case d i m (V 3) = 3 of a conjecture of the second author about the exterior operad Λ V d S 2 . For this we introduce a collection of natural involutions on the set of homogeneous cycle-free d-partitions of the complete graph K 2 d , and show that these involutions correspond to the relations in Λ V d S 2 (2 d + 1) . When d = 3 this correspondence allows us to give an explicit description of a determinant-like map and to settle the above mentioned conjecture. [ABSTRACT FROM AUTHOR]

In this article, we are interested in some subspaces of the exterior algebra of a simple Lie algebra g . In particular, we prove that some graded subspaces of degree d generate the g -module ∧ d g for some integers d. [ABSTRACT FROM AUTHOR]

This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated. [ABSTRACT FROM AUTHOR]

As an application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra \mathfrak {g} with a non-simply laced Dynkin diagram. We prove that in type B, C or F these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of \mathfrak {g}. [ABSTRACT FROM AUTHOR]

The goal of this paper is to supply an explicit description of the universal factorization algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r , as a representation of Lie algebras of n × n matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra g l ∞ due to Date, Jimbo, Kashiwara and Miwa. [ABSTRACT FROM AUTHOR]

In the paper we propose a proof of Reeder's Conjecture on the graded multiplicities of small representation in the exterior algebra Λ g for simple Lie algebras of type B and C. [ABSTRACT FROM AUTHOR]

Let K be a field, E the exterior algebra of a finite dimensional K-vector space, and F a finitely generated graded free E-module with homogeneous basis g1, . . ., gr such that deg g1 ≤ deg g2 ≤ · · · ≤ deg gr. We characterize the Hilbert functions of graded E–modules of the type F/M, with M graded submodule of F. The existence of a unique lexicographic submodule of F with the same Hilbert function as M plays a crucial role.

Assume that r is a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, p is a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and ν : r → o s p (p) is a homomorphism of Lie superalgebras. In this paper, we give a necessary and sufficient condition for r ⊕ p to be a quadratic Lie superalgebra. Then, we define the cubic Dirac operator D (g , r) on g and give a formula of (D (g , r)) 2 . Finally, we get the Vogan's conjecture for quadratic Lie superalgebras by D (g , r) . [ABSTRACT FROM AUTHOR]

Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate polynomials are definite determinantals. Indeed, in general, it is a difficult problem to decide whether a given polynomial is definite determinantal, and if it is, it is of paramount interest to determine a definite determinantal representation of that polynomial. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree d. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. [ABSTRACT FROM AUTHOR]

We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally complete intersection ring are self-dual under Grothendieck duality. This was proved by Stevenson when the ring is a quotient of a regular ring modulo a regular sequence; we offer two independent proofs in the more general setting. Second, we use these techniques to supply new proofs that complete intersections possess symmetry of complexity. [ABSTRACT FROM AUTHOR]

Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann and Clifford algebras and to the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics. [ABSTRACT FROM AUTHOR]

We define a map from second quandle homology to the Schur multiplier and examine its properties. Furthermore, we express the second homology of Alexander quandles in terms of exterior algebras. Additionally, we present a self-contained proof of its structure and provide some computational examples. [ABSTRACT FROM AUTHOR]

For C r (r > 1) random dynamical systems F on a compact smooth Riemannian manifold M , the fiber topological entropy is bounded above by an integral formula. Particularly, for the C ∞ random dynamical systems, the integral formula coincides with the fiber topological entropy. [ABSTRACT FROM AUTHOR]

Let K be a field, E the exterior algebra of a finite dimensional K -vector space, and F a finitely generated graded free E -module with homogeneous basis g2,...,gr such that degg1 = degg2 = ··· = deggr. Given the Hilbert function of a graded E--module of the type F/M, with M graded submodule of F, the existence of the unique lexicographic submodule of F with the same Hilbert function as M is proved by a new algorithmic approach. Such an approach allows us to establish a criterion for determining if a sequence of nonnegative integers defines the Hilbert function of a quotient of a free E-- module only via the combinatorial Kruskal--Katona's theorem. [ABSTRACT FROM AUTHOR]

The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of G r iso (3 , 7) or G r iso (4 , 8). [ABSTRACT FROM AUTHOR]

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.

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms. [ABSTRACT FROM AUTHOR]

In this paper we prove the concavity of the k -trace functions, A ↦ (Tr k [ exp ⁡ (H + ln ⁡ A) ]) 1 / k , on the convex cone of all positive definite matrices. Tr k [ A ] denotes the k -th elementary symmetric polynomial of the eigenvalues of A. As an application, we use the concavity of these k -trace functions to derive tail bounds and expectation estimates on the sum of the k largest (or smallest) eigenvalues of a sum of random matrices. [ABSTRACT FROM AUTHOR]

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in [ n ] which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank n by constructing such combinatorial systems for odd n. These constructions provide counterexamples to conjectures made by Domoskos and Zubor. [ABSTRACT FROM AUTHOR]

Abstract We give a lower bound on the Hilbert series of the exterior algebra modulo a principal ideal generated by a generic form of odd degree and disprove a conjecture by Moreno-Socías and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases. [ABSTRACT FROM AUTHOR]

A ring R with center C is said to be centrally essential if the module RC is an essential extension of the module CC. We describe centrally essential exterior algebras of finitely generated free modules over not necessary commutative rings and study properties of semi-Artinian centrally essential rings. [ABSTRACT FROM AUTHOR]

A new lower bound for the maximal length of a multivector is obtained. It is much closer to the best known upper bound than previously reported lower bound estimates. The maximal length appears to be unexpectedly large for n-vectors, with n>2, since the few exactly known values seem to grow only linearly with vector space dimension, whereas the new lower bound grows at power n-1 like the best known upper bound. This result has implications in quantum chemistry for the compression of information contained in an electronic wave function. [ABSTRACT FROM AUTHOR]

Highlights • The theory of screws is undergoing a revival in machines and robotics. • The theory is significantly generalized and extended in various directions. • Use is made of exterior algebra of multi-vectors. • A formulation in terms of tensor densities is also provided. Abstract The theory of screws is generalized to yield a formulation of Statics in higher-dimensional spaces in the absence of any distinguished metric structure. This level of generality necessitates the use of the tools of the exterior algebra of multi-vectors. The minimal structure retained serves to bring into relief the essential difference between screws and counter-screws as well as a weak version of the notion of rigid-body velocity field. An essential feature of the theory is the introduction of higher-order forces and displacements, whose relevance in the context of structured media can be determined in specific potential applications. [ABSTRACT FROM AUTHOR]

We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not always. We introduce two spectral sequences which help to explain the relationship between these cohomology groups. As a sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups. [ABSTRACT FROM AUTHOR]

We introduce a general class of second-order boundary value problems unifying application areas such as acoustics, electromagnetism, elastodynamics, quantum mechanics, and so on, into a single framework. This also enables us to solve wave propagation problems very efficiently with a single software system. The solution method precisely follows the conservation laws in finite-dimensional systems, whereas the constitutive relations are imposed approximately. We employ discrete exterior calculus for the spatial discretization, use natural crystal structures for three-dimensional meshing, and derive a "discrete Hodge" adapted to harmonic wave. The numerical experiments indicate that the cumulative pollution error can be practically eliminated in the case of harmonic wave problems. The restrictions following from the CFL condition can be bypassed with a local time-stepping scheme. The computational savings are at least one order of magnitude. [ABSTRACT FROM AUTHOR]

We study colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group S n acts on the multilinear components of these algebras. While S n acts trivially on the multilinear components of the colored symmetric algebra, we use poset topology techniques to understand the representation on its Koszul dual. We introduce an S n -poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are S n -isomorphic to the top cohomology modules of its maximal intervals. We use a technique of Sundaram to compute group representations on Cohen–Macaulay posets to give a generating formula for the Frobenius series of the colored exterior algebra. We exploit that formula to find an explicit expression for the expansion of the corresponding representations in terms of irreducible S n -representations. We show that the two colored Koszul dual algebras are Koszul in the sense of Priddy. [ABSTRACT FROM AUTHOR]

Let (x 1 , ... , x n , y 1 , ... , y n) be a list of 2 n commuting variables, (θ 1 , ... , θ n , ξ 1 , ... , ξ n) be a list of 2 n anticommuting variables, and C [ x n , y n ] ⊗ ∧ { θ n , ξ n } be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded S n -isomorphism type of C [ x n , y n ] ⊗ ∧ { θ n , ξ n } / I where I is the ideal generated by S n -invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables x i , y i equal to zero. [ABSTRACT FROM AUTHOR]

The basic concepts of exterior calculus for space−time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space−time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms.

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra gl m | n ( C ) . The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac–Moody Lie superalgebra gl m | n ˆ ( C ) on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter μ , and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter μ is nonzero. [ABSTRACT FROM AUTHOR]

In this article we define a number of varieties and sets in the projective space $${\mathbb{P} \left(\wedge^{2}\mathbb{R}^{n}\right)}$$ , which are obtained from the spectral analysis of a 2-tensor in $${\wedge^2\mathbb{R}^n}$$ . We refer to these sets as $${\mathcal{G}_{\mathcal{V}}}$$ , special cases of which are the Grassmannian and its extremal variety, as well as other sets. We study the problem of finding the projective variety whose 2-tensors in $${\wedge^{2}\mathbb{R}^{n}}$$ have the maximum distance from these varieties, via the use of a special mapping which we call Extr $${(\cdot)}$$ . It is shown that successive applications of this mapping lead either to the Grassmannian or its Extremal variety. [ABSTRACT FROM AUTHOR]

In [1], the authors computed the additive and multiplicative structure ofHH*(A;A), whereAis then-th exterior algebra over a field. In this paper, we derive all their results using a different method (AMT) as well as calculate the additive structure ofHHk(A;A) andHHk(A;A) overℤ. We provide concise presentations of algebrasHH∗(A;A) andHH*(A;A) as well as determine their generators in the Hochschild complex. Finally, we compute an explicit free resolution (spanned by multisets) of theAe-moduleAand describe the homotopy equivalence to its bar resolution. [ABSTRACT FROM PUBLISHER]

Given a3-vectorthe least distance problem from the Grassmann varietyis considered. The solution of this problem is related to a decomposition ofinto a sum of at most five decomposable orthogonal 3-vectors in. This decomposition implies a certain canonical structure for the Grassmann matrix which is a special matrix related to the decomposability properties of. This special structure implies the reduction of the problem to a considerably lower dimension tensor space ⊗3R2where the reduced least distance problem can be solved efficiently. [ABSTRACT FROM PUBLISHER]