Let $A$ be a commutative Noetherian ring, and let $R = A[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $A$. Let ${\mathfrak S}_{X}$ be the group of permutations of $X$. The group ${\mathfrak S}_{X}$ acts on $R$ in a natural way, and this in turn gives $R$ the structure of a left module over the left group ring $R[{\mathfrak S}_{X}]$. We prove that all ideals of $R$ invariant under the action of ${\mathfrak S}_{X}$ are finitely generated as $R[{\mathfrak S}_{X}]$-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.
We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
Symmetric strongly shifted ideals are a class of monomial ideals which come equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper we focus on algebraic and combinatorial properties of symmetric strongly shifted ideals. On the algebraic side, we elucidate properties that pertain to behavior under ideal operations, primary decomposition, and the structure of their Rees algebra. On the combinatorial side, we develop a notion of partition Borel generators which leads to connections to discrete polymatroids, conv…
A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gröbner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$.
Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and Shimonozo expand upon this, finding the reduced Groebner basis of the ideal of elementary symmetric polynomials in $n$ variables of degree $d$ for $d=n-k+1,\dots,n$ for $k\leq n$. In this paper, we further generalize their findings by using symbolic computation and experimentation to construct the reduced Groebner basis for the ideal generated by the elemen…
Recently, chains of increasing symmetric ideals have attracted considerable attention. In this note, we summarize some results and open problems concerning the asymptotic behavior of several algebraic and homological invariants along such chains, including codimension, projective dimension, Castelnuovo-Mumford regularity, and Betti tables.
Let $K$ be a field, and let $R = K[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $K$. Let ${\mathfrak S}_{X}$ be the symmetric group of $X$. The group ${\mathfrak S}_{X}$ acts naturally on $R$, and this in turn gives $R$ the structure of a left module over the (left) group ring $R[{\mathfrak S}_{X}]$. A recent theorem of Aschenbrenner and Hillar states that the module $R$ is Noetherian. We prove that submodules of $R$ can have any number of minimal generators.
We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a monomial radical if their coefficients are sufficiently general with respect to their supports. We prove this for instance in the case where some generator contains a power of a variable. Moreover, if the polynomials have only square-free terms and their coefficients do not sum to zero, then in a larger polynomial ring the ideal itself is square-free monomial. …
In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.
It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have finite Gröbner bases up to symmetry for certain monomial orders. An algorithm is presented for computing equivariant Gröbner bases that terminates whenever a finite basis exists, improving on previous algorithms that only guaranteed termination in rings Noetherian up to symmetry. This algorithm can be used to compute equivariant Gröbner bases of the above tori…
We study algebraic and homological properties of the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all $(n-1)$-minors of a symmetric $n \times n$ matrix whose entries in the upper triangle are distinct variables or zeros and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded as a simple graph $G$ with $n$ vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute al…
We define a set of invariants of a homogeneous ideal $I$ in a polynomial ring called the symmetric iterated Betti numbers of $I$. For $I_Γ$, the Stanley-Reisner ideal of a simplicial complex $Γ$, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of $Γ$ introduced by Duval and Rose. We show that the symmetric iterated Betti numbers of an ideal $I$ coincide with those of a particular reverse lexicographic generic initial ideal $\Gin(I)$ of $I$, and interpret these invariants in terms of the associated primes and standard pairs of $\Gin(I)$. We verify that for an…
We define a family of ideals $I_h$ in the polynomial ring $\mathbb{Z}[x_1,...,x_n]$ that are parametrized by Hessenberg functions $h$ (equivalently Dyck paths or ample partitions). The ideals $I_h$ generalize algebraically a family of ideals called the Tanisaki ideal, which is used in a geometric construction of permutation representations called Springer theory. To define $I_h$, we use polynomials in a proper subset of the variables ${x_1,...,x_n}$ that are symmetric under the corresponding permutation subgroup. We call these polynomials {\em truncated symmetric functions} and show combinator…
We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of Nevo who determined the exterior algebraic shifted complex of two disjoint simplicial complexes as predicted by Kalai.
We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.
Let $R=\mathbf{C}[ξ_1,ξ_2,\ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $\mathfrak{S}$. We classify the $\mathfrak{S}$-primes of $R$, determine the containments among these ideals, and describe the equivariant spectrum of $R$. We emphasize that $\mathfrak{S}$-prime ideals need not be radical, which is a primary source of difficulty. Our results yield a classification of $\mathfrak{S}$-ideals of $R$ up to copotency. Our work is motivated by the interest and applications of $\mathfrak{S}$-ideals seen in recent years.
Let $R$ be the polynomial ring $K[x_{i,j}]$ where $1 \le i \le r$ and $j \in \mathbb{N}$, and let $I$ be an ideal of $R$ stable under the natural action of the infinite symmetric group $S_{\infty}$. Nagel--Römer recently defined a Hilbert series $H_I(s,t)$ of $I$ and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.
We propose the notion of GAS numerical semigroup which generalizes both almost symmetric and 2-AGL numerical semigroups. Moreover, we introduce the concept of almost canonical ideal which generalizes the notion of canonical ideal in the same way almost symmetric numerical semigroups generalize symmetric ones. We prove that a numerical semigroup with maximal ideal $M$ and multiplicity $e$ is GAS if and only if $M-e$ is an almost canonical ideal of $M-M$. This generalizes a result of Barucci about almost symmetric semigroups and a theorem of Chau, Goto, Kumashiro, and Matsuoka about 2-AGL semigr…
Let $R =k[x_1, \cdots, x_n,\cdots]$ be the infinite variable polynomial ring equipped with the natural $\mathfrak{S}_\infty$ action, where $k$ is a field of characteristic zero. In recent work \cite{NS21}, Nagpal--Snowden gave an indirect proof that $\mathfrak{S}_\infty$-ideal generated by $(x_1-x_2)^{2n}$ is not $\mathfrak{S}_\infty$-prime. In this paper, we give a direct proof, with explicit elements. We further formulate some conjectures on possible generalizations of the result.
Given a representation of a finite group $G$ over some commutative base ring $\mathbf{k}$, the cofixed space is the largest quotient of the representation on which the group acts trivially. If $G$ acts by $\mathbf{k}$-algebra automorphisms, then the cofixed space is a module over the ring of $G$-invariants. When the order of $G$ is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that $G$ is the symmetric group on $n$ letters acting on a polynomial ring by permuting its variables. When $\mathbf{k}$ has characteristic 0, the co…
Given an ascending chain $(I_n)_{n\in\mathbb{N}}$ of $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals $(I_n^\vee)_{n\in\mathbb{N}}$. Using a novel combinatorial tool, which we call \emph{avoidance up to symmetry}, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of $I_n^\vee$ is given by a polynomial in $n$ for sufficiently large $n$. The same is true for the num…
Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with $a \in A$ generate a prime ideal in S or the set of polynomials $p_a$ with $a \in A$ is a regular sequence in S. We produce a large families of prime ideals by exploiting Serre's criterion for normality [4, Theorem 18.15] with the help of arithmetic considerations, vanishing sums of roots of unity [9]. We also deduce several other results concerning regular …
The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.
Matrix Schubert varieties are the closures of the orbits of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many othe…
In this paper we develop a Grobner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain ther…
In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major algorithm is most performant over finite fields whereas the second algorithm is a probabilistic modification of the modular computation of Groebner bases based on the articles by Arnold (cf. [A03]), Idrees, Pfister, Steidel (cf. [IPS11]) and Noro, Yokoyama (cf. [NY12], [Y12]). In fact, the first algorithm that mainly uses the given symmetry, improves the necessary…
We investigate the quotient ring $R$ of the ring of formal power series $\Q[[x_1,x_2,...]]$ over the closure of the ideal generated by non-constant quasi-\break symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from $(0,0)$ and above the line $y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in $n$ variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of $R_n$ is bounded above by t…
Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of $t$ linear ideals is at most $t$ and they also showed…
We show that the apolar ideal to the determinant of a generic symmetric matrix is generated in degree two, and the apolar ideal to the permanent of a generic symmetric matrix is generated in degrees two and three. In each case we specify the generators of the apolar ideal. As a consequence, using a result of K. Ranestad and F. O. Schreyer we give lower bounds to the cactus rank and rank of each of these polynomials. We compare these bounds with those obtained by J. Landsberg and Z. Teitler.
In this article, we study monomial curves, toric ideals and monomial algebras associated to $4$-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen-Macaulay.
Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 that are biratio…
We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.
We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory.
This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V)xGL(W)-invariant ideal in the symmetric algebra S of the tensor product of V with the dual of W, where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal. Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we will compute the test ideals (and so …
We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.
It is shown that the toric ideal of the centrally symmetric configuration of the order polytope of a finite partially ordered set possesses a squarefree quadratic initial ideal. It then follows that the convex polytope arising from the centrally symmetric configuration of an order polytope is a normal Gorenstein Fano polytope.
Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of non-minimal generators in a module. We also obtain a simple proof and an extension of a result by Kwieciński, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.
We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the…
We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2.
Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded version…
We provide a locally free resolution of the projectivized symmetric algebra of the ideal sheaf of a zero-dimensional scheme defined by n + 1 equations in an n-dimensional variety. The resolution is given in terms of the resolution of the ideal itself and of the Eagon-Northcott complex of the Koszul hull.
