We enumerate the number of isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p2. To do this, we enumerate GL (2, p)-orbits of sets of irreducible, monic polynomials in 𝔽p[x]. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

The main purpose of this paper is to obtain an explicit Capelli identity relating skew-symmetric matrices under the action of the general linear group $GL_N$. In particular, we give an explicit formula for the skew Capelli element in terms of the trace of powers of a matrix defined by the standard infinitesimal generators of $GL_N$.

AMS 2000 Mathematics subject classification: Primary 17B35. Secondary 17B10; 15A15; 20G05

We formulate a generalization of Givental–Kim's quantum hyperplane principle. This is applied to compute the quantum cohomology of a Calabi–Yau 3-fold defined as the rank 4 locus of a general skew-symmetric 7×7 matrix with coefficients in P6. The computation verifies the mirror symmetry predictions of Rødland [25].