Let \(K\) be a field, and let \(X= (X_{ij})\) be a matrix of indeterminates. A subset \(Y\) of \(X\) is called a ladder if \(X_{ik}\), \(X_{hj}\) belong to \(Y\) whenever \(X_{ij}, X_{hk}\in Y\) and \(i< h\) and \(j< k\). One defines \(I_t (Y)\) to be the ideal generated by the \(t\)-minors of \(X\) which involve only indeterminates of \(Y\). The ring \(R_t (Y)= K[ Y]/ I_t (Y)\) is called a ladder determinantal ring.
Ladder determinantal rings were introduced by S. Abhyankar [Enumerative combinatorics of Young tableaux, Marcel Dekker, New Nork (1988; Zbl 0643.05001)] in his studies of singularities of Schubert varieties of flag manifolds. By results of H. Narasimhan [J. Algebra 102, 162-185 (1986; Zbl 0604.14045)], J. Herzog and Ngo Viet Trung [Adv. Math. 96, No. 1, 1-37 (1992; Zbl 0778.13022)] and A. Conca [J. Pure Appl. Algebra 98, No. 2, 119-134 (1995; Zbl 0842.13007)], ladder determinantal rings are known to be normal Cohen-Macaulay domains.
The aim of the paper is to characterize the Gorenstein property of a ladder determinantal ring \(R_t (Y)\) in terms of the shape of the ladder \(Y\). This has already been done for \(t=2\) by T. Hibi [in Commutative algebra and combinatorics, US-Jap. joint Semin., Kyoto 1985, Adv. Stud. Pure Math. 11, 93-109 (1987; Zbl 0654.13015)] and for determinantal rings associated with one-sided ladders by A. Conca (loc. cit.).
In order to treat the general case, the author determines the divisor class group and the canonical class of \(R_t (Y)\). This is done by showing first that a localization \(R_t (Y) [f^{-1}]\) of \(R_t (Y)\) is factorial. It follows from the Nagata theorem that the divisor class group is generated by the minimal primes of \(f\). The minimal primes of \(f\) are determined by means of Gröbner bases arguments. Then it turns out that the divisor class group \(\text{Cl} (R_t (Y))\) of \(R_t (Y)\) is free and it has a basis which is in some sense the natural one. Let \(\text{cl} (P_i)\) denote the elements of this basis. An inductive argument allows one to determine the representation \(\text{cl} (\omega)= \sum \lambda_i \text{cl} (P_i)\) of the canonical class \(\text{cl} (\omega)\) of \(R_t (Y)\) with respect to the basis of \(\text{Cl}(R_t(Y)\). To do this one has to overcome the problem that the uniqueness of the canonical module can be destroyed by arbitrary localizations. The coefficients \(\lambda_i\) are expressed in terms of the coordinates of the so-called inside corners of \(Y\). It follows that \(R_t (Y)\) is Gorenstein if and only if all the \(\lambda_i\) vanish, and this is the case if and only if the inside corners of the ladder lie on certain diagonals.