Hodge Algebras Research Articles

Quantifying the Structural Stability of Simplicial Homology

Simplicial complexes are generalizations of classical graphs. Their homology groups are widely used to characterize the structure and the topology of data in e.g. chemistry, neuroscience, and transportation networks. In this work we assume we are given a simplicial complex and that we can act on its underlying graph, formed by the set of 1-simplices, and we investigate the stability of its homology with respect to perturbations of the edges in such graph. Precisely, exploiting the isomorphism between the homology groups and the higher-order Laplacian operators, we propose a numerical method to compute the smallest graph perturbation sufficient to change the dimension of the simplex’s Hodge homology. Our approach is based on a matrix nearness problem formulated as a matrix differential equation, which requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra’s homology groups. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method’s performance on a variety of synthetic quasi-triangulation datasets and real-world transportation networks.

Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties

The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi ideals by associating an ideal with every interpolating polytope. This is a family of polytopes that generalizes the order and chain polytopes of a poset (à la Fang–Fourier–Litza–Pegel). We then describe a maximal cone in the Gröbner fan of each of these ideals. We also establish some useful facts concerning PBW-semistandardness, in particular, we prove that it provides a new Hodge algebra structure on the Plücker algebra.

A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.

Chevalley groups of polynomial rings over Dedekind domains

Abstract Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥ 2 {\geq 2} . We prove that G ⁢ ( R ⁢ [ x 1 , … , x n ] ) = G ⁢ ( R ) ⁢ E ⁢ ( R ⁢ [ x 1 , … , x n ] ) for any ⁢ n ≥ 1 . G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G = SL N , Sp 2 ⁢ N {G=\mathrm{SL}_{N},\mathrm{Sp}_{2N}} . We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

Isotropic reductive groups over discrete Hodge algebras

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank $$\ge n$$ , if every normal semisimple reductive R-subgroup of G contains $$({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n$$ . We prove that if G has isotropic rank $$\ge 1$$ and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra $$A=R[x_1,\ldots ,x_n]/I$$ over R, the map $$H^1_{\mathrm {Nis}}(A,G)\rightarrow H^1_{\mathrm {Nis}}(R,G)$$ induced by evaluation at $$x_1=\cdots =x_n=0$$ , is a bijection. If k has characteristic 0, then, moreover, the map $$H^1_{\acute{\mathrm{e}}\mathrm {t}}(A,G)\rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(R,G)$$ has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is $$\ge 2$$ , and A is square-free, then $$K_1^G(A)=K_1^G(R)$$ , where $$K_1^G(R)=G(R)/E(R)$$ is the corresponding non-stable $$K_1$$ -functor, also called the Whitehead group of G. The corresponding statements for $$G={{\mathrm{GL}}}_n$$ were previously proved by Ton Vorst.

Hibi Algebras and Representation Theory

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

Projective modules and orbit space of unimodular rows over Discrete Hodge algebras over a non-Noetherian ring

For any commutative ring $R$ of Krull dimension zero and for any discrete Hodge algebra $D$ over $R$, it is proven that, if $n\geq 3$, the group $E_n(D)$ of $n\times n$ elementary matrices acts transitively on $Um_n(D)$, the set of unimodular rows of length $n$ over $D$.

From standard monomial theory to semi-toric degenerations via Newton–Okounkov bodies

The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper we construct these semi-toric degenerations using quasi-valuations and triangulations of Newton-Okounkov bodies.

Efficient generation of ideals in a discrete Hodge algebra

Let R be a commutative Noetherian ring and D be a discrete Hodge algebra over R of dimension d>dim(R). Then we show that(i) the top Euler class group Ed(D) of D is trivial,(ii) if d>dim(R)+1, then (d−1)-st Euler class group Ed−1(D) of D is trivial.

Augmented Superfield Approach to Nilpotent Symmetries of the Modified Version of 2D Proca Theory

We derive the complete set of off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, and (anti-)co-BRST symmetry transformations forallthe fields of the modified version of two(1+1)-dimensional (2D) Proca theory by exploiting the “augmented” superfield formalism where the (dual-)horizontality conditions and (dual-)gauge invariant restrictions are exploitedtogether. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian density in the language of superfield approach. We also express the nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST charges within the framework of augmented superfield formalism. This exercise leads to somenovelobservations which have, hitherto, not been pointed out in the literature within the framework of superfield approach to BRST formalism. For the sake of completeness, we also mention, very briefly, a unique bosonic symmetry, the ghost-scale symmetry, and discrete symmetries of the theory and show that the algebra of conserved charges provides a physical realization of the Hodge algebra (satisfied by the de Rham cohomological operators of differential geometry).

The isomorphism problem for commutative monoid rings

By substantial changes and corrections in Demushkin's old paper the essentially final positive answer to the isomorphism problem for monoid rings of submonoids of Z r is obtained. This means that the underlying monoid is shown to be determined (up to isomorphism) by the corresponding monoid ring. Thereafter the positive answer to the analogous question for the ‘dual’ objects — descrete Hodge algebras — is derived.

Action of some braid groups on hodge algebras

In this paper we show that the braid groups B n and the symmetric automorphism groups H(n) of the free group F n,n = 3,4 act in a non-linear way on an algebra with straightening law (ordinal Hodge algebra). We indicate various properties of these rings.

Hodge algebra structures on certain rings of invariants and applications

Let B be a commutative ring with identity, m, n, and r be positive integers such that r ≤ min{m, n}, a 1, …, a r (resp. b 1, … b r ) be integers such that 1 ≤ a 1< … < a r ≤ m (resp. 1 ≤ b 1 < … < b r < n) and U (resp. V) be the most general m × r (resp. r × n) matrix such that s-minors of first a s − 1 rows (resp. b s − 1 columns) of U (resp. V) are all zero for s = 1, …, r. We investigate the B-algebra C generated by all the entries of UV and all the r-minors of U and V. We introduce a Hodge algebra structure, to which the discrete Hodge algebra associate is Cohen Macaulay, on C and prove that C is Cohen-Macaulay if so is B. Using this Hodge algebra structure, we show that C is the ring of absolute invariants of a certain group action, compute the divisor class group and the canonical class of C, and give a criterion of Gorenstein property of C in terms of a 1 ,…, a r and b 1…, b r .

Homogeneous complete intersection Hodge algebras on simplicial complexes

Homogeneous complete intersection Hodge algebras on simplicial complexes

Some properties of projective modules over discrete Hodge algebras

All rings in this paper are assumed to be commutative with 1 and all modules finitely generated. Let B be a discrete Hodge algebra over a.r ing A. By definition this means that B=R/I where R=A[XI ..... X,,] and I is an ideal in R generated by monomials. T. Vorst showed in 1-12] that thebehavior of projective modules over such B is very similar to the case of an ordinary polynomial ring in the following sense: If all projective modules over any polynomial ring with ground ring A are, extended from A, then every projective module over a discrete Hodge algebra over A is extended from A. Let B be a discrete Hodge algebra over a Noetherian ring A and P a projective module over B. S. Mandal proved in [7] that if rank P~> dim B/> dim A + 1 then

A remark on Hodge algebras and Gröbner bases

A remark on Hodge algebras and Gröbner bases

Blow-up of Straightening-Closed Ideals in Ordinal Hodge Algebras

We study a class of ideals $I$ in graded ordinal Hodge algebras $A$. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for $A/{I^n}$ which, under certain conditions, is sharp for $n$ large. Frequently one observes that ${I^n}= {I^{(n)}}$. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".

Blow-up of straightening-closed ideals in ordinal Hodge algebras

We study a class of ideals I I in graded ordinal Hodge algebras A A . These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for A / I n A/{I^n} which, under certain conditions, is sharp for n n large. Frequently one observes that I n = I ( n ) {I^n}= {I^{(n)}} . Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".

On the presentation of hodge algebras and the existence of hodge algebra structures

On the presentation of hodge algebras and the existence of hodge algebra structures

On the variety of projectors

In this paper we are going to give an answer to the following two questions. Let K be a field and A4 the space of n x n square matrices with coefficients in K. The first question we want to investigate is what are the equations of the subvariety A’, of ikf, consisting of rank h matrices A such that A2 = A. By equations we mean of course a set of explicit generators of the prime ideal of x,. The second question is what are the equations of the varieties Yh, h <n/2, which are the closures of the nilpotent conjugacy classes in h4, E, consisting of matrices of rank h whose square is the matrix 0. In this case we shall also prove that these varieties Y, are normal and Cohen-Macaulay in arbitrary characteristic, thus generalizing a result of Hesselink [3] to arbitrary characteristic. Note that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and Cohen-Macaulay by Kraft and Procesi [4], but our method is not suitable for generalizing their result. The method used in this paper is the one of Hodge algebras [ 11, namely we shall exhibit an explicit linear basis for the coordinate rings of the varieties X,, and Y,.