Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector

Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where V =\gfq^n and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in \GL_n(\gfq). In fact, we determine \gfq[V \oplus V^*]^G for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of "doubly parametrized" series of group actions whose invariant rings are known to have a uniform description.Comment: 16 page

The 2-matrix of the spin-polarized electron gas: contraction sum rules and spectral resolutions

The spin-polarized homogeneous electron gas with densities $\rho_\uparrow$ and $\rho_\downarrow$ for electrons with spin `up' ($\uparrow$) and spin `down' ($\downarrow$), respectively, is systematically analyzed with respect to its lowest-order reduced densities and density matrices and their mutual relations. The three 2-body reduced density matrices $\gamma_{\uparrow\uparrow}$, $\gamma_{\downarrow\downarrow}$, $\gamma_a$ are 4-point functions for electron pairs with spins $\uparrow\uparrow$, $\downarrow\downarrow$, and antiparallel, respectively. From them, three functions $G_{\uparrow\uparrow}(x,y)$, $G_{\downarrow\downarrow}(x,y)$, $G_a(x,y)$, depending on only two variables, are derived. These functions contain not only the pair densities but also the 1-body reduced density matrices. The contraction properties of the 2-body reduced density matrices lead to three sum rules to be obeyed by the three key functions $G_{ss}$, $G_a$. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions $n_\uparrow(k)$ and $n_\downarrow(k)$, following from $f_\uparrow(r)$ and $f_\downarrow(r)$ by Fourier transform, are correctly normalized through $f_s(0)=1$. In addition to the non-negativity conditions $n_s(k),g_{ss}(r),g_a(r)\geq 0$ [these quantities are probabilities], it holds $n_s(k)\leq 1$ and $g_{ss}(0)=0$ due to the Pauli principle and $g_a(0)\leq 1$ due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin-unpolarized homogeneous electron gas in terms of 2-body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin-polarized case and (ii) to the 2-body reduced density matrix giving thus its spectral resolutions.Comment: 32 pages, 4 figure