Winfried Bruns, Wolmer V. Vasconcelos
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.
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In this article we consider the exterior power and the symmetric tensors of the polynomial ring in one variable. The structure of an associative semigraded algebra of this polynomial ring induces on the symmetric tensors the structure of an associative semigraded algebra, and on the exterior power induces structure of a semigraded module over semigraded algebra of symmetric tensors. The algebra of symmetric polynomials is isomorphic to the algebra of the symmetric tensors of polynomial ring in one variables. We obtained the explicit expression for symmetric polynomials via elementary symmetric…
We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order.
We establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the number of facets, and that the exterior shift is the best among the exterior shifting operations in the sense that it increases the number of facets the least. Methods of proof include Gröbner basis theory over the exterior algebra, Cartan homology, degree functions, and Alexander duality.
Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain $R$. By exploiting a connection with symmetric determinantal representations, we characterize the image of the principal minor map through the condition that certain polynomials coming from so-called Rayleigh differences are squares in the polynomial ring over $R$. In almost all cases, one can characterize the image of the principal minor map using the orbit of Cayley's hyperdeterminant under the action of $(SL_2(R))^{n} \rtimes S_{n}$. Over the complex numbers, this recove…
We study the exterior depth of an $E$-module and its exterior generic annihilator numbers. For the exterior depth of a squarefree $E$-module we show how it relates to the symmetric depth of the corresponding $S$-module and classify those simplicial complexes having a particular exterior depth in terms of their exterior shifting. We define exterior annihilator numbers analogously to the annihilator numbers over the polynomial ring introduced by Trung and Conca, Herzog and Hibi. In addition to a combinatorial interpretation of the annihilator numbers we show how they are related to the symmetric…
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 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.
In the present paper, exterior algebraic shifting and symmetric algebraic shifting of bipartite graphs and chordal graphs are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that, for $a>3$ and $b>3$, the exterior algebraic shifted graph of the complete bipartite graph $K_{a,b}$ of size $a,b$ is different from the symmetric algebraic shifted graph of $K_{a,b}$. Second, we will show that the exterior algebraic shifted graph of any chordal graph $G$ is coincident with the symmetric algebraic shifted graph of $G$.
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…
The principal minors of a symmetric $n{\times}n$-matrix form a vector of length $2^n$. We characterize these vectors in terms of algebraic equations derived from the $ 2{\times}2{\times}2$-hyperdeterminant.
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 determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m x n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.
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 consider a family of schemes, that are defined by minors of a homogeneous symmetric matrix with polynomial entries. We assume that they have maximal possible codimension, given the size of the matrix and of the minors that define them. We show that these schemes are G-bilinked to a linear variety of the same dimension. In particular, they can be obtained from a linear variety by a finite sequence of ascending G-biliaisons on some determinantal schemes. In particular, it follows that these schemes are glicci. We describe the biliaisons explicitely in the proof of the main theorem.
Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.
We describe noncommutative desingularizations of determinantal varieties, determinantal varieties defined by minors of generic symmetric matrices, and pfaffian varieties defined by pfaffians of generic anti-symmetric matrices. For maximal minors of square matrices and symmetric matrices, this gives a non-commutative crepant resolution. Along the way, we describe a method to calculate the quiver with relations for any non-commutative desingularizations coming from exceptional collections over partial flag varieties.
Let S_n denote the symmetric group on n letters. We consider the S_n-root lattice A_{n-1} = {(z1,...,zn) in Z^n | z1+...+zn = 0}, where S_n acts on Z^n by permuting the coordinates, and its tensor, symmetric, and exterior squares. For odd values of n, we show that the tensor square is equivalent, in the sense of Colliot-Thelene and Sansuc, to the exterior square. Consequently, the rationality problem for generic division algebras, for odd values of n, amounts to proving stable rationality of the multiplicative S_n-invariant field of the exterior square of A_{n-1}. Furthermore, confirming a con…
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