Action of $\mathfrak{osp}(1|2n)$ on polynomials tensor ${\mathbb{C}}^{0|2n}$

Abstract

For each positive integer $n$, the basic classical complex Lie superalgebra $\mathfrak{osp}(1|2n)$ has a unique equivalence class of infinite-dimensional completely-pointed modules, those weight modules with one-dimensional weight spaces. The polynomials $\mathbb{C}[x_1,x_2,\dots ,x_n]$ in $n$ indeterminates is a choice representative. In the case $n>1$, tensoring polynomials with the natural $\mathfrak{osp}(1|2n)$-module ${\mathbb{C}}^{1|2n}$ gives rise to a tensor product representation $V=\mathbb{C}[x_1,x_2,\dots ,x_n]\otimes {\mathbb{C}}^{1|2n}$ of $\mathfrak{osp}(1|2n)$ that decomposes into two irreducible summands. These summands are understood through automorphisms of $V$ that we determine as intertwining operators describing the first summand as an isomorphic copy of $\mathbb{C}[x_1,x_2,\dots ,x_n]$ and the second summand as $\mathbb{C}[x_1,x_2,\dots ,x_n]\otimes {\mathbb{C}}^{2n}$, which does not have a natural $\mathfrak{osp}(1|2n)$-module structure and is not a paraboson Fock space with known bases. We present the intertwining operators as infinite diagonal block matrices of arrowhead matrices and give bases, along with formulas for the action of the odd root vectors (which generate $\mathfrak{osp}(1|2n)$) on these bases, for each of these conjugated oscillator realizations. We also revisit an expected difference: The decomposition of $\mathbb{C}[x]\otimes {\mathbb{C}}^{1|2}$ yields three irreducible summands instead of two.

21 pages