KRS and determinantal ideals (Q2717163)

Let \(X=(x_{ij})\) be a \(m\times n\) matrix with indeterminate coefficients, \(R=K[X]\) be the ring of polynomials of \(x_{ij}\) over a field \(K\). First of all, the authors recall the main notions and ideas of the following works: \textit{W. Bruns} and \textit{U. Vetter}, ``Determinantal rings'', Lect. Notes Math. 1327 (1988; Zbl 0673.13006), \textit{W. Bruns} and \textit{A. Conca}, Compos. Math. 111, No. 1, 111-122 (1998; Zbl 0937.05082) and \textit{W. Bruns} and \textit{M. Kwiecinski}, Math. Z. 233, No. 1, 155-126 (2000; Zbl 1081.13502). In particular, according to the Doubilet-Rota-Stein straightening law, the \(K\)-vector space basis of \(K[X]\) can be formed by standard products of minors (*) \(\Sigma=\delta_1\cdot\dots\cdot\delta_\omega\) (where \(\delta_1, \dots, \delta_\omega\) are ordered in a special way, the shape of \(\Sigma\) being the descending sequence of sizes of minors \(|\delta_1 |, \dots, |\delta_\omega |)\).NEWLINENEWLINENEWLINEThe authors consider mainly the ideals of \(K[X]\) having the last property (ideals with a standard basis). The Knuth-Robinson-Schensted transformation KRS \((\Sigma)\) establishes a bijective correspondence between the standard products (*) and monomials of \(K[X]\) [``KRS correspondence'', see, for instance, \textit{W. Fulton}, ``Young tableau'' (1997; Zbl 0878.14034) and \textit{R. P. Stanley}, ``Enumerative combinatorics''. I (1999; Zbl 0945.05008) and II (2001; Zbl 0978.05002)]. NEWLINENEWLINENEWLINEThe ideals \(I\) are investigated from the point of view of obtaining their initial ideals and Gröbner bases by means of this correspondence [the first result in this direction was obtained by \textit{B. Sturmfels}, Math. Z. 205, No. 1, 137-144 (1990; Zbl 0685.13005)] for the ideals \(I_t\), that are generated by all minors of the size \(t)\). The notion of a KRS-invariant function \(F:D\to N\) (where \(D\) is the set of all products of minors) is introduced and that allows one to find many ideals admitting the required possibility. In particular, the functions \(\alpha_k\) of \textit{C. Greene} [Adv. Math. 14, 254-265 (1974; Zbl 0303.05006)] and \(\gamma_t\) of \textit{C. de Concini}, \textit{D. Eisenbud} and \textit{C. Procesi} [Invent. Math. 56, 129-165 (1980; Zbl 0435.14015)] are KRS-invariants. A connection between them is investigated. The authors consider also a new function, related to the ideal \(I(X,\delta)\), cogenerated by a minor \(\delta\), but they can not prove their KRS-invariance. Besides, a class of ideals, defined by shape (that are characterized, in particular, in the case \(\text{char} K=0\), by the property of being stable under the action of the group \(GL(m,K)\times GL(n,K))\) is investigated and results from the above mentioned point of view are obtained. Results of the work of \textit{W. Bruns} and \textit{A. Conca} (loc. cit.) concerning symbolic powers of ideals are complemented.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].