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Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by $\mathfrak {R}(A)=\oplus _{(g,h)\in G\times H}A_{(g,h)}\otimes R_{h}$ . In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a, $b\in \mathbb {N}$ , we find a basis for the graded identities and the graded central polynomials for the algebra Ma,b(E), graded by the group $G\times \mathbb {Z}_{2}$ . Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural $\mathbb {Z}_{2}$ -grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group $G\times \mathbb {Z}_{2}$ , so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on Ma,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product Ma,b(E) ⊗ Mr,s(E) is PI-equivalent to Mar+bs,as+br(E). Finally, when the grading group is $\mathbb {Z}_{3}\times \mathbb {Z}_{2}$ (resp. $\mathbb {Z}\times \mathbb {Z}_{2}$ ), we present a complete description of a basis for the graded central polynomials for the algebra M2,1(E) (resp. Ma,b(E) in the case b = 1).

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).

We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogues of local and global LYM inequalities, the Erd\H{o}s-Ko-Rado theorem, and the Ahlswede-Khachatrian bound for $t$-intersecting hypergraphs. Using these tools, we prove a new extension of the Two Families Theorem of Bollob\'{a}s, giving a weighted bound for subspace configurations satisfying a skew cross-intersection condition. We also verify a recent conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patk\'{o}s, Tompkins, and Xiao on pairs of set systems satisfying both an intersection and a cross-intersection condition., Comment: 29 pages, substantial revision: new results in sections 2.5 (exterior downwards local LYM), 2.6 (exterior Ahslwede-Khachatrian bound), and 5 (proof of a conjecture of Gerbner et al, arxiv:1910.04666[math.CO])

In the paper, we continue the study of the relatively free Grassmann algebra $${\mathbb{F}^{(3)}}$$ without unit over an infinite field of characteristic 2, which was initiated in previous works of the author. The main attention is paid here to the relationship between the $$\mathbb{T}$$-spaces of n-words, i.e., the $$\mathbb{T}$$-spaces generated by all monomials in $${\mathbb{F}^{(3)}}$$ containing each of their variables with multiplicity n. The results of this note will enable one to form a more complete picture of possible inclusions between the $$\mathbb{T}$$-spaces of r-and n-words for r > n.

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional $$({\mathcal {G}}, \mu , \gamma , \phi )$$, correspond a unique Lie algebra structure on $${\mathcal {D}}= {\mathcal {G}}\oplus \mathcal {G^{*}}$$ which leaves invariant the canonical scalar product on $${\mathcal {D}}$$, called the double of the given Lie quasi-bialgebra. We show that there exist on $$\Lambda {\mathcal {G}}$$, the exterior algebra of $${\mathcal {G}}$$, a $${\mathcal {D}}$$-module structure.

Let $$G=GL(m|n)$$ be the general linear supergroup over an algebraically closed field K of characteristic zero, and let $$G_{ev}=GL(m)\times GL(n)$$ be its even subsupergroup. The induced supermodule $$H^0_G(\lambda )$$, corresponding to a dominant weight $$\lambda $$ of G, can be represented as $$H^0_{G_{ev}}(\lambda )\otimes \Lambda (Y)$$, where $$Y=V_m^*\otimes V_n$$ is a tensor product of the dual of the natural GL(m)-module $$V_m$$ and the natural GL(n)-module $$V_n$$, and $$\Lambda (Y)$$ is the exterior algebra of Y. For a dominant weight $$\lambda $$ of G, we construct explicit $$G_{ev}$$-primitive vectors in $$H^0_G(\lambda )$$. Related to this, we give explicit formulas for $$G_{ev}$$-primitive vectors of the supermodules $$H^0_{G_{ev}}(\lambda )\otimes \otimes ^k Y$$. Finally, we describe a basis of $$G_{ev}$$-primitive vectors in the largest polynomial subsupermodule $$\nabla (\lambda )$$ of $$H^0_G(\lambda )$$ (and therefore in the costandard supermodule of the corresponding Schur superalgebra S(m|n)). This yields a description of a basis of $$G_{ev}$$-primitive vectors in arbitrary induced supermodule $$H^0_G(\lambda )$$.

We translate into double forms formalism the basic Greub and Greub-Vanstone identities that were previously obtained in mixed exterior algebras. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; we show that the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms. We define and study a refinement of the notion of pure curvature of Maillot, namely p " role="presentation"> p p p -pure curvature, and we use one of the basic identities to prove that if a Riemannian n " role="presentation"> n n n -manifold has k " role="presentation"> k k k -pure curvature and n ≥ 4 k " role="presentation"> n ≥ 4 k n ≥ 4 k n\geq 4k then its Pontryagin class of degree 4 k " role="presentation"> 4 k 4 k 4k vanishes.

In this paper, we consider the exterior algebra $\Lambda(W)$ of a polynomial $\mathrm{GL}(n)$-module $W$ and use previously developed methods to determine the Hilbert series of the algebra of invariants $\Lambda(W)^G$, where $G$ is one of the classical complex subgroups of $\mathrm{GL}(n)$, namely $\mathrm{SL}(n)$, $\mathrm{O}(n)$, $\mathrm{SO}(n)$, or $\mathrm{Sp}(2d)$ (for $n=2d$). Since $\Lambda(W)^G$ is finite dimensional, we apply the described method to compute a lot of explicit examples. For $\Lambda(S^3\mathbb{C}^3)^{\mathrm{SL}(3)}$, using the computed Hilbert series, we obtain an explicit set of generators.

In the paper we propose a proof of Reeder’s Conjecture on the graded multiplicities of small representation in the exterior algebra ${\Lambda } \mathfrak {g}$ for simple Lie algebras of type B and C.

Let { ( A i , B i ) } i = 1 m be a collection of pairs of sets with | A i | = a and | B i | = b for 1 ≤ i ≤ m . Suppose that A i ∩ B j = 0 if and only if i = j , then by the famous Two Families Theorem of Bollobas, we have the size of this collection m ≤ a + b a . In this paper, we consider a variant of this problem by requiring { A i } i = 1 m to be intersecting additionally. Using exterior algebra method, we prove a weighted Bollobas-type theorem for finite dimensional real vector spaces under these constraints. As a consequence, we obtain a similar theorem for finite sets, which settles a recent conjecture of Gerbner et al. (2020). Moreover, we also determine the unique extremal structure of { ( A i , B i ) } i = 1 m for the primary case of the theorem for finite sets.

We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper., Comment: 9 pages

In this work, we study the classification of differential graded algebras over $\mathbb{Z}$ (DGAs) whose homology is $\mathbb{F}_p[x]$, i.e. the polynomial algebra over $\mathbb{F}_p$ on a single generator. This classification problem was left open in work of Dwyer, Greenlees and Iyengar. For $\lvert y_{2p-2} \rvert = 2p-2$, we show that there is a unique non-formal DGA with homology $\mathbb{F}_p[y_{2p-2}]$ and a non-formal $2p-2$ Postnikov section. Among a classification result, this provides the first example of a non-formal DGA with homology $\mathbb{F}_p[x]$. By duality, this also shows that there is a non-formal DGA whose homology is an exterior algebra over $\mathbb{F}_p$ with a generator in degree $-(2p-1)$. Considering the classification of the ring spectra corresponding to these DGAs, we show that every $E_2$ DGA with homology $\mathbb{F}_p[x]$ (with no restrictions on $\lvert x \rvert$) is topologically equivalent to the formal DGA with homology $\mathbb{F}_p[x]$, i.e. they are topologically formal. This follows by a theorem of Hopkins and Mahowald., Minor revisions

We propose a generalization of the parallelogram identity in any dimension ?$N\geq 2$?, establishing the ratio of the quadratic mean of the diagonals to the quadratic mean of the faces of a parallelotope. The proof makes use of simple properties of the exterior product of vectors. Predstavimo posplošitev paralelogramske identitete v poljubni dimenziji ?$N\geq 2$?, s tem da določimo razmerje kvadratne sredine diagonal in kvadratne sredine lic paralelotopa. Dokaz uporablja enostavne lastnosti zunanjega produkta vektorjev.

Let p be an odd prime number, F a field of characteristic zero, and let Ebe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector space L. We determine the bases of the ℤp-graded identities.Moreover we compute the ℤp-graded codimension and cocharacter sequences for the algebra E endowed with any ℤp-grading such that L is a homogeneous subspace.

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.

The generalized wreath product of permutation groups was introduced by Evdokimov and Ponomarenko in order to study the schurity problem for S-rings over cyclic groups. In this paper we construct the generalized wreath product of permutation groups by a method entirely different from Evdokimov and Ponomarenko’s construction. Then we give a necessary and sufficient condition for the wedge product of schurian association schemes coming from the generalized wreath product of permutation groups.

We show that the codimension sequence of the algebra of k × k matrices over the Grassmann algebra, c n ( M k ( E ) ) , is asymptotic to α n 1 − k 2 2 ( 2 k 2 ) n , where α is an undetermined constant.

Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,\dots ,f_c\in S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements. The homotopies for the $f_{i}$ on a free resolution of $M$ induce a natural structure of graded $E$-module on ${\rm Tor}^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},\dots,f_{c})$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$. Turning to ${\rm Ext}_{R}(M,\, k)$ we show that, when $M$ is a high syzygy over $R$, the minimal free resolution of ${\rm Ext}_{R}(M,\, k)$ as a module over the ring of CI operators is the Bernstein-Gel'fand-Gel'fand dual of the $E$-module ${\rm Tor}^{S}(M,\,k)$. For the proof we introduce \emph{higher CI operators}, and give a construction of a (generally non-minimal) resolution of $M$ over $S$ starting from a resolution of $M$ over $R$ and its higher CI operators., To appear in the Journal of the European Mathematical Society (JEMS)

We devise an algorithm that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ± ε. Our algorithm runs in time ε−2 4k(n+m) poly(k). The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic 2k·poly(n) time algorithm to find a k-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect k-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.

A quadratic algebra A is said to be a Frobenius algebra (or Frobenius quantum space) of dimension d if (a) \(\dim A_d=1\), \(A_i=0\) for \(i>d\). For all j, the multiplication map \(m:A_j\otimes A_{d-j} \rightarrow A_d\) is a perfect duality. (For example where this pairing is nonsymmetric for \(j=d-j\), see [30]. This asymmetry is the reason why the quantum determinant considered in Example 9.6 might be noncentral.) The algebra A is called a quantum Grassmann algebra if, in addition, (c) \(\dim A_i = \left( {\begin{array}{c}d\\ i\end{array}}\right) \).

After recalling the definition of Grassmann algebra and elements of Grassmann--Berezin calculus, we use the expression of Pfaffians as Grassmann integrals to generalize a series of formulas relating generating functions of paths in digraphs to Pfaffians. We start with the celebrated Lindstr\"om-Gessel-Viennot formula, which we derive in the general case of a graph with cycles. We then make further use of Grassmann algebraic tools to prove a generalization of the results of (Stembridge 1990). Our results, which are applicable to graphs with cycles, are formulated in terms of systems of nonintersecting paths and nonintersecting cycles in digraphs., Comment: 12 pages

We start with the commutative polynomial algebra $$\begin{aligned} A=\mathbb {K}\langle \tilde{x}_1,\dots ,\tilde{x}_n\rangle /(\tilde{x}_i\tilde{x}_j - \tilde{x}_j\tilde{x}_i)\;. \end{aligned}$$

The study of orthogonal multiplications is more than 100 years old and goes back to the works of Hurwitz and Radon. Yet, apart from the extensive literature on admissibility of domain and range dimensions near the Hurwitz-Radon range (in codimension \(\le 8\)), only sporadic and fragmentary results are known about full classification (in large codimension), more specifically, about the moduli space \(\mathfrak {M}_m\) of orthogonal multiplications \(F:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}^n\) (for various n), even for \(m\le 4\). In this paper we give an insight to the subtle geometries of \(\mathfrak {M}_3\) and \(\mathfrak {M}_4\). Orthogonal multiplications are intimately connected to quadratic eigenmaps between spheres via the Hopf-Whitehead construction. The 9-dimensional moduli space \(\mathfrak {M}_3\) lies on the boundary of the 84-dimensional moduli of quadratic eigenmaps of \(S^5\) into spheres. Similarly, the 36-dimensional moduli space \(\mathfrak {M}_4\) is on the boundary of the 300-dimensional moduli of quadratic eigenmaps of \(S^7\) into spheres. We will show that \(\mathfrak {M}_3\) is the \(SO(3)\times SO(3)\)-orbit of a 3-dimensional convex body bounded by Cayley’s nodal cubic surface with vertices in a real projective space \(\mathbb {R}P^3\), the latter imbedded equivariantly and minimally in an 8-sphere of the space of quadratic spherical harmonics on \(S^3\). For \(\mathfrak {M}_4\), we show that it possesses two orthogonal 18-dimensional slices each of which is an \(SO(4)\times SO(4)\)-orbit of a 6-dimensional polytope \(\mathcal {P}\subset \mathbb {R}^6\). This polytope itself is the convex hull of two orthogonal regular tetrahedra. The corresponding orthogonal multiplications are explicitly constructible. Finally, we give an algebraic description of the 24-dimensional space of diagonalizable elements in \(\mathfrak {M}_4\). The crucial fact in \(\mathbb {R}^4\) is the splitting of the exterior product \(\Lambda ^2(\mathbb {R}^4)\) into self-dual and anti-self-dual components. The techniques employed here can be traced back to the work of Ziller and the author (Toth and Ziller 1999) in describing the 18-dimensional moduli for quartic spherical minimal immersions of \(S^3\) into spheres. As a new feature, we point out the importance of multiplicity of the zeros of the polynomials that define the boundary \(\partial \,\mathfrak {M}_m\) as a determinantal variety.

Rota’s basis conjecture states that in any square array of vectors whose rows are bases of a fixed vector space the vectors can be rearranged within their rows in such a way that afterwards not only the rows are bases, but also the columns. We discuss an online version of this conjecture, in which the permutation used for rearranging the vectors in a given row must be determined without knowledge of the vectors further down the array. The paper contains surprises both for those who believe this online basis conjecture at first glance, and for those who disbelieve it. Keywords: Rota’s basis conjecture; Exterior algebra; Online algorithms

We study the shifted analogue of the "Lie--Poisson" construction for $L_\infty$ algebroids and we prove that any $L_\infty$ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair $(L,A)$, the space $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ admits a degree $(+1)$ derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley--Eilenberg differential $d_A^{\operatorname{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\to \Omega^{\bullet +1}_A(\Lambda^\bullet(L/A))$ as unary $L_\infty$ bracket. This degree $(+1)$ derived Poisson algebra structure on $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley--Eilenberg hypercohomology $\mathbb{H}(\Omega^{\bullet}_A(\Lambda^\bullet(L/A)),d_A^{\operatorname{Bott}})$ admits a canonical Gerstenhaber algebra structure., Comment: 37 pages

Let F be a field of characteristic zero and let E be the unitary Grassmann algebra generated by an infinite-dimensional F-vector space L. Denote by E=E(0)⊕E(1) an arbitrary Z2-grading on E such that the subspace L is homogeneous. Given a superalgebra A=A(0)⊕A(1), define its generalized Grassmann envelope A⊗E as the superalgebra A⊗E=(A(0)⊗E(0))⊕(A(1)⊗E(1)). Note that when E is the canonical grading of E then A⊗E is the Grassmann envelope of A. In this work we describe the generators of the T2-ideal, Idgr(UTk,l(F)⊗E), of the Z2-graded polynomial identities of the superalgebras UTk,l(F)⊗E, as well as linear bases of the corresponding relatively free graded algebras. Here, given k⩾1, l⩾0, UTk,l(F) is the algebra of (k+l)×(k+l) upper triangular matrices over F with the Z2-grading UTk+l(F)=(UTk(F)00UTl(F))⊕(0Mk×l(F)00). In order to prove our result we obtain a similar description corresponding to the T-ideals Id(UTn(E)) and Id(UTn(Gr)) of ordinary polynomial identities, where Gr is the Grassmann algebra generated by an r-dimensional vector space.

An eight-vertex model on a square lattice over a Grassmann algebra is investigated using an equation that is a three-dimensional generalization of the Yang-Baxter equation. Anticommuting quantum spin systems are studied, where the quasiclassical limit leads to some abstract classical physics with anticommuting variables. The solution of the quantum Yang-Baxter equation is the R - matrix, which corresponds to the transfer R - matrix of the eight-vertex model of statistical mechanics.

In 1994 Dias da Silva and Hamidoune solved a long-standing open problem of Erd?s and Heilbronn using the structure of cyclic spaces for derivatives on Grassmannians and the representation theory of symmetric groups. They proved that for any subset A of the p -element group Z / p Z (where p is a prime), at least min { p , m | A | - m 2 + 1 } different elements of the group can be written as the sum of m different elements of A . In this note we present an easily accessible simplified version of their proof for the case m = 2 , and explain how the method can be applied to obtain the corresponding inverse theorem.

In this paper, we present a function called the Determinant-Like Function that generalises the determinant function to m by n matrices using the Clifford algebra Cl(V, 0). By definition, this generalisation satisfies the property that the exterior product of its column vectors has a magnitude that equals its determinant-like function if it has more rows than columns. For matrices which have more columns than rows, we make use of another property that the exterior product of its rows has a magnitude that equals the absolute value of its determinant-like function if it has more columns than rows. Defining the sign of this determinant-like function remains an open question.

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition., Comment: 84 pages

The classic Cayley identity states that \det(\partial) (\det X)^s = s(s+1)...(s+n-1) (\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and \partial=(\partial/\partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-parametrized" Cayley identities, and the "product-parametrized" and "border-parametrized" rectangular Cayley identities., Comment: LaTeX2e, 144 pages. Version 2 has a slightly changed title and abstract, and includes new remarks in Sections 1, 2.6 and (especially) 9 concerning prehomogeneous vector spaces and citing the prior work of Sugiyama (2011). To be published in Advances in Applied Mathematics

Let $X$ be a compact K\"ahler manifold of dimension $n.$ Let $T$ and $S$ be two positive closed currents on $X$ of bidegree $(p,p)$ and $(q,q)$ respectively with $p+q\le n.$ Assume that $T$ has a continuous super-potential. We prove that the wedge product $T \wedge S,$ defined by Dinh and Sibony, is a positive closed current., Comment: 11 pages

Given a (Figure presented.)3-vector (Formula presented.) the least distance problem from the Grassmann variety (Formula presented.) is considered. The solution of this problem is related to a decomposition of (Formula presented.) into a sum of at most five decomposable orthogonal 3-vectors in (Formula presented.). This decomposition implies a certain canonical structure for the Grassmann matrix which is a special matrix related to the decomposability properties of (Formula presented.). This special structure implies the reduction of the problem to a considerably lower dimension tensor space ⊗3R2 where the reduced least distance problem can be solved efficiently.

Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. To a given monomial ideal $I\subsetneq E$ we associate a special monomial ideal $J$ with generators in the same degrees as those of $I$ and such that the number of the minimal monomial generators in each degree of $I$ and $J$ coincide. We call $J$ the colexsegment ideal associated to $I$. We prove that the class of strongly stable ideals in $E$ generated in one degree satisfies the colex lower bound, that is, the total Betti numbers of the colexsegment ideal associated to a strongly stable ideal $I\subsetneq E$ generated in one degree are smaller than or equal to those of $I$., To appear in Pure and Applied Mathematics Quarterly

Let F be the free unitary associative algebra over a field F on the set X = {x_1, x_2, ...}. A vector subspace V of F is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F. A T-subspace V in F is limit if every larger T-subspace W \gneqq V is finitely generated (as a T-subspace) but V itself is not. Recently Brand\~ao Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p>2 the T-subspace C(G) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F is unique, that is, there are no limit T-subspaces in F other than C(G). In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F over an infinite field F of characteristic p>2. For each k \ge 1, the limit T-subspace R_k arises from the central polynomials in 2k variables of the Grassmann algebra G., Comment: 22 pages

Let g \mathfrak g be a finite-dimensional complex semisimple Lie algebra and b \mathfrak b a Borel subalgebra. Then g \mathfrak g acts on its exterior algebra ∧ g \wedge \mathfrak g naturally. We prove that the maximal eigenvalue of the Casimir operator on ∧ g \wedge \mathfrak g is one third of the dimension of g \mathfrak g , that the maximal eigenvalue m i m_i of the Casimir operator on ∧ i g \wedge ^i\mathfrak g is increasing for 0 ≤ i ≤ r 0\le i\le r , where r r is the number of positive roots, and that the corresponding eigenspace M i M_i is a multiplicity-free g \mathfrak g -module whose highest weight vectors correspond to certain ad-nilpotent ideals of b \mathfrak {b} . We also obtain a result describing the set of weights of the irreducible representation of g \mathfrak g with highest weight a multiple of ρ \rho , where ρ \rho is one half the sum of positive roots.

We investigate necessary conditions for the existence of projections of polytopes that preserve full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of R��rig & Ziegler (2009) and complement their results., 18 pages, 2 figures

Let F be an infinite field of characteristic p>2 and let E be the Grassmann algebra generated by an infinite dimensional vector space V over F. In this paper, we describe the T2-ideal of the Z2-graded polynomial identities of the Grassmann algebra E for any Z2-grading such that V is homogeneous in the grading. In particular, we give a description of the T2-ideal of the graded identities of E in the case there is a finite number of homogeneous elements of the linear basis of E belonging to one of the homogenous components of E.

Let K be a field of characteristic 0, and let UTn(K) be the algebra of n × n upper triangular matrices over K. We denote by Pn the vector space of all multilinear polynomials of degree n in x1, …, xn in the free associative algebra K(X). Then Pn is a left Sn-module where the symmetric group Sn acts on Pn by permuting the variables. The Sn-modules Pn and KSn are canonically isomorphic, a fact that lets us employ the representation theory in the study of algebras with polynomial identities. Denote by An the alternative subgroup of Sn. One may study KAn and its isomorphic copy in Pn with the corresponding action of An. Henke and Regev studied A-identities. They described the A-codimensions of the Grassmann algebra and conjectured a finite generating set of the A-identities for E. In an earlier paper we answered in the affirmative their conjecture. Another problem posed by Henke and Regev concerned the minimal degree of an A-identity satisfied by the full matrix algebra Mn(K). They asked whether this minimal degree equals 2n+2. In this paper we show that this is not the case as long as n ≥ 6. Our main result consists in computing a lower bound for the minimal degree d(n) of an A-identity satisfied by the algebra UTn(K). It turns out that, given any positive integer k, there exists n0 such that d(n) > 2n + k for all n > n0. Moreover, we compute d(n) for n ≤ 4 and for n = 6. It turns out that d(6) = 15 > 2 × 6 + 2. We have reasons to believe that for every n, d(n) = [(5n + 1)/2] holds, where as usual [x] stands for the integer part of the real number x, that is the largest integer ≤ x.

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that An-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1 ⩽ k ⩽ n − 1). With each hyperplane H of An-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧n−k V) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of An-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A5,3(F).

Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ℤ2-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A(0) ⊕ A(1), define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ℤ2-graded identities and we describe the ℤ2-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT2(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading.

In this paper we describe the central polynomials for the infinite-dimensional unitary Grassmann algebra G over an infinite field F of characteristic ≠ 2. We exhibit a set of polynomials that generates the vector space C(G) of the central polynomials of G as a T-space. Using a deep result of Shchigolev we prove that if charF = p > 2 then the T-space C(G) is not finitely generated. Moreover, over such a field F, C(G) is a limit T-space, that is, C(G) is not a finitely generated T-space but every larger T-space W ≩ C(G) is. We obtain similar results for the infinite-dimensional nonunitary Grassmann algebra H as well.

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces ``of unusually large genus'' that first appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and Wills (1983). Via ``projections of deformed wedge products'' we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in R^3. As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (``moduli'') for the surfaces in R^3. In order to prove that there are many moduli, we introduce the concept of ``affine support sets'' in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in R^3 via dual 4-polytopes., Comment: 17 pages, 5 figures

Let be the regular representation of a finite abelian group G and let ${\mathcal{C}}_{n}$ denote the cyclic group of order n. For $G={\mathcal{C}}_{n}$ , we compute the Poincare series of all ${\mathcal{C}}_{n}$ -isotypic components in (the symmetric tensor exterior algebra of ). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili---Jibladze. Then we consider the Cayley table, , of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of equals , where n is the order of G.

Let m and k be two fixed positive integers such that m > k ⩾ 2 . Let V be a left vector space over a division ring with dimension at least m + k + 1 . Let G m ( V ) be the Grassmannian consisting of all m -dimensional subspaces of V . We characterize surjective mappings T from G m ( V ) onto itself such that for any A , B in G m ( V ) , the distance between A and B is not greater than k if and only if the distance between T ( A ) and T ( B ) is not greater than k .

We give details of the 1-1 correspondence between equiangular frames of n vectors for R d and graphs with n vertices. This has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory. The nontight examples given here (which correspond to graphs with more than 2 eigenvalues) have the potential for similar applications, e.g., the frame corresponding to the 5-cycle graph is the unique Grassmannian frame of 5 vectors in openR 3 . Further, the associated canonical tight frames have a small number of angles in many cases.

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \({\mathbb {Z}^n}\)-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.