Finite Dimensional K-vector Spaces Research Articles

Mapping cones of monomial ideals over exterior algebras

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.

The Space of Traces in Symmetric Monoidal Infinity Categories

Abstract We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(𝟙,𝟙)$ for all dualizable objects x in any symmetric monoidal $\infty$-category $\mathtt{C}$. This generalizes the trace from linear algebra that assigns a scalar $\operatorname{Tr}(\,f\,) \in k$ to any endomorphism f : V → V of a finite-dimensional k-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence, we show that the trace $\operatorname{Tr}$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterizations of the $\infty$-categorical trace. By restricting the aforementioned notion of tracelike transformations from endomorphisms to automorphisms one can in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto \operatorname{Tr}(\,f^{\,n})$. Unlike for $\operatorname{Tr}$, the relevant connected component of the moduli space is not contractible, but rather equivalent to $B\mathbb{Z}/n\mathbb{Z}$ or BS1 for n = 0. As a result, we obtain a $\mathbb{Z}/n\mathbb{Z}$-action on $\operatorname{Tr}(\,f^{\,n})$ as well as a circle action on $\operatorname{Tr}(\operatorname{id}_x)$.

A generalization of Kruskal–Katona’s theorem

Abstract 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 g 1, . . ., gr such that deg g 1 ≤ deg g 2 ≤ · · · ≤ 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.

The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten theory

In this work, for a finite group G and a 4-cocycle ω∈Z4(G,k×), we compute explicitly the center of the monoidal 2-category 2VecGω of ω-twisted G-graded 1-categories of finite dimensional k-vector spaces. This center gives a precise mathematical description of the topological defects in the associated 3+1D Dijkgraaf-Witten TQFT. We prove that this center is a braided monoidal 2-category with a trivial sylleptic center.

A right-invariant lattice-order on groups of paraunitary matrices

In [14], Rump defined and characterized noncommutative universal groups G(X) for generalized orthomodular lattices X.We give an explicit description of G(X) in terms of paraunitary matrix groups, whenever X is the orthomodular lattice of subspaces of a finite-dimensional k-vector space V that is equipped with an anisotropic, symmetric k-bilinear form.

The finite unipotent groups consisting of bireflections

Abstract Let k be a field of characteristic p and V a finite-dimensional k-vector space. An element g ∈ GL ⁢ ( V ) {g\in{\rm GL}(V)} is called a bireflection if it centralizes a subspace of codimension less than or equal to 2. It is known by a result of Kemper that if for a finite p-group G ≤ GL ⁢ ( V ) {G\leq{\rm GL}(V)} the ring of invariants Sym ⁢ ( V * ) G {{\rm Sym}(V^{*})^{G}} is Cohen–Macaulay, G is generated by bireflections. Although the converse is false in general, it holds in special cases e.g. for particular families of groups consisting of bireflections. In this paper we give, for p > 2 {p>2} , a classification of all finite unipotent subgroups of GL ⁢ ( V ) {{\rm GL}(V)} consisting of bireflections. Our description of the groups is given explicitly in terms useful for exploring the corresponding rings of invariants. This further analysis will be the topic of a forthcoming paper.

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.

On Ranks of Polynomials

Let V be a vector space over a field k, P : V → k, d ≥ 3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V and a polynomial P : V → k of degree d such that rank(∂P/∂t) ≤ r for all t ∈ V − 0. Our proof of this theorem is based on the application of results on Gowers norms for finite fields k. We don’t know a direct proof even in the case when k = ℂ.

Structure of certain Weyl modules for the Spin groups

Let K be an algebraically closed field of characteristic p⩾0 and let W be a finite-dimensional K-vector space of dimension greater than or equal to 5. In this paper, we give the structure of certain Weyl modules for G=Spin(W) in the case where p≠2, as well as the dimension of the corresponding irreducible, finite-dimensional, rational KG-modules. In addition, we determine the composition factors of the restriction of certain irreducible, finite-dimensional, rational KSL(W)-modules to SO(W).

Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products

Let k be a local field of characteristic 0, and let G be a connected semisimple almost k-algebraic group. Suppose rankkG⩾1 and ρ is an excellent representation of G on a finite dimensional k-vector space V. We construct uniform pointwise bounds for the K-finite matrix coefficients restricted on G of all unitary representations of the semi-direct product G⋉ρV without non-trivial V-fixed vectors. These bounds turn out to be sharper than the bounds obtained from G itself for some cases. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of the pair (G⋉ρV,V).

Rank functions on rooted tree quivers

The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with a unique sink), then the ring $R(Q)_{red}$ is a finitely generated $\Z$-module (here $R(Q)_{red}$ is the ring R(Q) modulo the ideal of all nilpotent elements). We will describe the ring $R(Q)_{red}$ explicitly, by studying functors from the category Rep(Q) of representations of Q over K to the category of finite dimensional K-vector spaces. We also present an open problem for future direction.

Bimonads and Hopf monads on categories

AbstractThe purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on : we define a bimonad on as an endofunctor B which is a monad and a comonad with an entwining λ : BB → BB satisfying certain conditions. This λ is also employed to define the category of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ : BB → BB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining λ and lead to an extension of the theory of braided Hopf algebras.An antipode is defined as a natural transformation S : B → B with special properties. For categories with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between and the category of B-bimodules. This is a general form of the Fundamental Theorem of Hopf algebras.Finally we observe a nice symmetry: If B is an endofunctor with a right adjoint R, then B is a (Hopf) bimonad if and only if R is a (Hopf) bimonad. Thus a k-vector space H is a Hopf algebra if and only if Homk(H,−) is a Hopf bimonad. This provides a rich source for Hopf monads not defined by tensor products and generalises the well-known fact that a finite dimensional k-vector space H is a Hopf algebra if and only if its dual H* = Homk(H,k) is a Hopf algebra. Moreover, we obtain that any set G is a group if and only if the functor Map(G,−) is a Hopf monad on the category of sets.

Lie Algebras with a Coalgebra Splitting

In their recent article [5], the authors endow every finite-dimensional simple complex Lie algebra g with a coalgebra structure such that the composition μ◦δ of the two structure maps δ : g −→ g⊗Cg and μ : g⊗C g −→ g coincides with the identity. Moreover, the dual algebra (g∗, δ∗) associated to the Lie coalgebra is isomorphic to (g, μ). The coalgebra map δ is given explicitly for sl(n), those for the other types are obtained via embeddings g ↪→ sl(n). The purpose of the present short note is to elicit the conceptual sources of [5], starting from the observation that the coalgebra maps defined in [5] are in fact homomorphisms of g-modules. For Lie algebras affording non-degenerate symmetric associative forms, such coalgebra maps naturally arise by dualizing the multiplication. This immediately implies the abovementioned duality, and the formulae displayed in [5, §4] can also be subsumed under our general approach. By demanding that δ be g-linear, we depart from the usual compatibility condition of a Lie bialgebra, which requires δ to be a derivation. In that case, μ ◦ δ is a derivation of g. For fields of characteristic zero, only nilpotent Lie algebras afford invertible derivations (cf. [7]), so that non-zero Lie bialgebras over such fields never satisfy μ ◦ δ = idg. Let k be a field. Given a finite-dimensional k-vector space V , we consider the k-linear maps

The Gauss-Manin connection and Tannaka duality

If f : X → Spec(K) is a smooth, geometrically connected variety defined over a field of characteristic 0, K ⊃ k is a field extension, and x ∈ X(K) is a rational point, one considers three Tannaka categories C(X/K), C(X/k), C(K/k) of flat connections with compatible fiber functors. The objects of C(X/K) are bundles (i.e., locally free coherent modules of finite type) with relative flat connections ((V,∇/K),∇/K : V → ΩX/K ⊗OX V), the ones of C(X/k) are bundles with flat absolute connections ((V,∇), ∇ : V → ΩX/k ⊗OX V), and the ones of C(K/k) are K-vector spaces with flat connections ((V,∇), ∇ : V → ΩK/k⊗KV). The morphisms are the flat morphisms and the fiber functor has values in the category of finite dimensional vector spaces VecK over K, defined by the restriction of V to x for C(X/K), C(X/k) and by V for C(K/k). Then C(X/K) is a neutral Tannaka category, and Tannaka duality [2,Theorem 2.11] yields the existence of a proalgebraic group schemeG(X/K) overK, so that C(X/K) becomes equivalent to the representation category Repf(G(X/K)) on finite dimensional K-vector spaces. The two other Tannaka categories C(X/k), C(K/k) are not necessarily neutral. We assume that they are defined over k, which is to say that k is the endomorphism ring EndC(K/k)((K, dK/k)) of the unit object,which in this case is the same as the subfield of K of flat sections. Equivalently, this is saying that k is algebraically closed in K. Then Tannaka duality [3, theoreme 1.12] yields the existence of groupoid schemes G(X/k), G(K/k) over k acting on Spec(K) ×k Spec(K), so that, in the groupoid sense, C(X/k) (resp., C(K/k)) becomes equivalent to the representation category Repf(K :

An algebraic proof of Zak's inequality for the dimension of the Gauss image

A classical theorem in complex algebraic geometry states that, for any smooth projective variety, the Gauss map is finite; in particular, a smooth variety and its Gauss image have the same dimension (with the obvious exception of a linear space). Furthermore, even when the variety is not smooth, Zak proved a lower bound on the dimension of its Gauss image in terms of the dimension of its singular locus. Our purpose in this note is to reinterpret Gauss maps within a more general algebraic framework, and thus recover Zak’s bound on the dimension of the Gauss image as a special case of an interesting new bound on the analytic spread of a module of Kahler differentials. This connects that classical subject in complex geometry to recent research in commutative algebra concerning integral closures of modules. In particular, we give a new, purely algebraic proof of Zak’s theorem. We recall a precise version of Zak’s theorem. Let X be an irreducible projective variety of dimension d defined over an algebraically closed field k, considered with a fixed embedding X ⊂ P(V ) for some finite dimensional k-vector space V . The Gauss map is the rational map from X to the Grassmannian of projective d-planes in P(V ) assigning to each smooth k-point of X the projective tangent plane there,

Orthogonal, symplectic and unitary representations of finite groups

Let K be a field, G a finite group, and ρ : G → GL(V ) a linear representation on the finite dimensional K-space V . The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms h : V × V → K which are G-invariant. II. If h is such a form, enumerate the equivalence classes of representations of G into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on G or K under which two orthogonal, symplectic or unitary representations of G are equivalent if and only if they are equivalent as linear representations and their underlying forms are “isotypically” equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) KG-module components of their spaces are equivalent. We assume throughout that the characteristic of K does not divide 2|G|. Solutions to I and II are given when K is a finite or local field, or when K is a global field and the representation is “split”. The results for III are strongest when the degrees of the absolutely irreducible representations of G are odd – for example if G has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index – and, in the case that K is a local or global field, when the representations are split. Let G be a finite group of order g, K a field of characteristic relatively prime to 2g with an involution (possibly the identity) a 7→ ā, and ρ : G → GL(V ) a linear representation of G on the finite dimensional K-vector space V . The group algebraKG has an involution extending that onK and inverting the group elements. Since KG is a semisimple algebra, the involution algebra (KG, ) is a direct sum of simple involution algebras: (KG, ) = (A1, )⊕ (A2, )⊕ · · · ⊕ (Ar, ), where each Ai either is a simple algebra stabilized by the involution or is a “simple hyperbolic algebra” – a direct sum of two simple algebras interchanged by the involution. Now let

A useful semistability criterion

We provide a short proof for a semistability criterion which is crucial to the construction of master spaces which has drawn interest in recent research in Geometric Invariant Theory. Among the main objects of study of Geometric Invariant Theory are representations of reductive algebraic groups. Given a reductive algebraic group G, defined over a field k, a finite dimensional k-vector space W, and a rational representation p: G GL(W), one obtains an action of G on IP(W), the space of hyperplanes in W, and a representation of G on the algebra R := Sym W. By the Hilbert-Nagata theorem, the algebra RG of G-invariant elements in R is a finitely generated graded subalgebra of R. Therefore, IP(W)//G Proj RG is a projective variety, and there is a rational map 7r: IP(W) --+ IP(W)//G. This is the GIT procedure of forming a quotient. A closed point [w] C EP(W) is called semistable, if 7r is defined in [w], i.e., if there is a non-constant homogeneous element in RG not vanishing in [w]. The set IP(W)88 of semistable points is open, and a point [w] c IP(W) is called polystable, if [w] is semistable and the G-orbit of [w] is closed in P(W)88. The map 7r identifies the closed points of IP(W)//G with the set of G-orbits of polystable points. A central task of GIT is now to identify the semistable and polystable points in IP(W). This is usually achieved by the Hilbert-Mumford criterion [1]. However, if p is the direct sum of other representations, this might become too difficult. In this note, we will provide a short and elementary proof of a theorem from [2], dealing with this situation. More precisely, let k and G be as before, and W1,...,W, finite dimensional k-vector spaces. Suppose we are given representations pi: G GL(Wi), i = 1, ..., s, with W = W1 ... ED W, and p = p, ED * * ffl For any t = (Li, ...,tt) with 0 0. In all these cases, one can safely apply the Hilbert-Mumford criterion. The outcome is that the semistable points correspond to semistable L-oriented pairs, using the formulation of semistability in terms of the parameter dependent semistability concept for Bradlow pairs. This content downloaded from 157.55.39.19 on Tue, 14 Jun 2016 05:30:35 UTC All use subject to http://about.jstor.org/terms A USEFUL SEMISTABILITY CRITERION 1925 PROOF OF THE THEOREM We first prove the semistable part. By definition, the point w is semistable if and only if, for some k > 0, there is an invariant section in H0((9(k)) = SkW which does not vanish in w. Now, SkW = Oki+*+k,=k Ski W0 (g ... S S . Ws and the representation of G on SkW respects this decomposition, i.e., (SkW)G (~kl+--+k =k(SkiWl (... (? SksWs)G. Therefore, w is semistable if and only if we find a section in some (Sk~l W,, o . Sklt Wt )G which does not vanish in w. This is precisely the assertion of the theorem. Note that we have used only the very definition of semistability. The polystable part will follow easily by means of an induction from the following ClaiR. Let w' ([w= 1,v WL2], [WL3], ..., [wjt]) be a point in P(WVV, e W142) X P(13 t. ) Then w' is semistable (polystable) w.r.t. the given linearization in ((k, k3, ..., kt), if and only if either ([WI, [wI 3 ],) ..., [wI J]) is semistable (polystable) in P(.tt) w.r.t. the linearization in O (k, k3I ...., kt) for either i = 1 (and w,2 = 0) or i = 2 (and W,, = 0), or there are positive natural numbers n, k1, and k2, such that k[ +-k2 = nk and the point (Q[W,], [W62], [WL3], ..., [wI ]W1) is semistable (polystable) in P(,1,2,,3.tt) w.r.t. the linearization in O (kj, k2, nk3, ..., nkt). The semistable part is proved as before. First, we suppose that w' is polystable. With the Hilbert-Mumford criterion [1], the claim can be rephrased as follows: There exist non-negative rational numbers K, and K2, not both zero, with fK1 + r/2= k such that for every one-parameter subgroup A of G, we have

K-Theory of Endomorphism Rings and of Rings of Invariants

This paper gives the following description ofK0of the endomorphism ring of a finitely generated projective module.Theorem.Let T be a ring and P a finitely generated,projective T-module. Let I be the trace ideal of P. Then K0(EndPT)is isomorphic to a subgroup of K0(T,I).If,further,the natural map K1(T)→K1(T/I)is surjective then K0(EndPT)is isomorphic to the subgroup of K0(T)generated by the direct summands of Pn,for n∈N.As a corollary we can determineK0of the ring of invariants for many free linear actions. In particular, the following result is proved.Theorem.Let V be a fixed-point-free linear representation of a finite group G over a field k of characteristic zero and let S(V)be the symmetric algebra of V.Let K be any finite-dimensional k-vector space. Then K0(S(V)G⊗kS(K))=〈[S(V)G⊗kS(K)]〉.Similar results are given for suitable noncommutative versions ofS(V).

Matrices, relations, and group representations

Motivated by the search for a good notion of a “selfdual” object in a symmetric monoidal category, we give an abstract notion of commutative separable algebra in a symmetric monoidal category V , such that when V is the category of modules over a commutative ring k we obtain the usual commutative separable algebras. Developing the calculus based on such an abstract notion we are able to prove that the dual category of separable algebras in any compact closed, additive category with coequalizers is in fact a boolean pretopos. We apply this result to give a simple characterization of the categories of continuous representations of profinite groups in discrete finite dimensional k-vector spaces. Finally we show that the same calculus can be applied to symmetric monoidal categories of relations to give an essentially algebraic characterization of such categories.

On open orbits and their complements

1.1. We fix an algebraically closed field k and a quiver Q without oriented cycles, whose vertex set Q, we identify with { 1, . . . . n}. For a in the set Q, of arrows of Q, we denote by ta and ha the tail and the head of a, respectively. A representation X of Q consists of a family {X(i): in QO} of finite dimensional k-vector spaces and a family {X(a): X(ta) + X(ha): a E Q1 } of k-linear maps. A morphism f: X + Y between two representations of Q is a family {f(i): X(i)+ Y(i): in QO} of k-linear maps satisfying f(ha)oX(a) = Y(a)of(ta) for all arrows a. We denote the category of representations of Q by mod Q. It is an abelian category of global dimension at most 1. We will denote the only possibly non-trivial extension group Ext’ by Ext. The dimension vector of a representation X of Q is the vector dim X= (dim X(l), . . . . dim X(n)) in N”, and the dimension of X is the natural number dim X= dim X( 1) + . . . + dim X(n). The support of X is the full subquiver of Q whose vertices are {i: X(i) ZO}. For a vector d = (d,, . . . . d,) E N”, we define R(Q, d) to be the set of representations X of Q such that X(i) = k” for ie QO. Thus R(Q, d) is the product R(Q, d) = n Hom,(kdru, kdhe).