Multilinear components of symmetric-minor and Pfaffian ideals

Let $k$ be a field, $t\ge 1$, and $0\le m<t$. Write $[t]=\{1,\dots ,t\}$ and let $e_1,\dots ,e_t$ be the standard basis of ${\mathbb{N}}^t$. In the polynomial ring on the generic symmetric $t\times t$ matrix $Z=(z_{ij})$, give $z_{ij}=z_{ji}$ multidegree $e_i+e_j$. For $I,J\subseteq [t]$ of size $m+1$, order each subset increasingly and write $Z_{I,J}$ for the corresponding minor. The multidegree $(1,\dots ,1)$ component of the ideal generated by all $(m+1)$-minors is spanned by $\det (Z_{I,J}){\prod }_{\{a,b\}\in \pi }{z}_{ab},$ where $I$ and $J$ are disjoint and $\pi$ is a perfect matching (a partition into two-element blocks) of $[t]\setminus (I\cup J)$. For a generic alternating $t\times t$ matrix $A=(a_{ij})$ with $a_{ji}=-a_{ij}$ and $a_{ii}=0$, give $a_{ij}$ multidegree $e_i+e_j$. Fix $1\le s\le \lfloor t\mathrm{/}2\rfloor$. For $S\subseteq [t]$, let $A_S$ be the principal submatrix with indices in increasing order. The multidegree $(1,\dots ,1)$ component of the ideal generated by principal Pfaffians of size $2s$ is spanned by $\mathrm{Pf}(A_S){\prod }_{\{a,b\}\in \pi }{a}_{ab},$ where $S=(s_1<\cdots <s_{2s})$ and $\pi$ is a perfect matching of $[t]\setminus S$. More precisely, for a perfect matching $M=\{\{i_1,j_1\},\dots ,\{i_s,j_s\}\}$ of $\{1,\dots ,2s\}$ (that is, a partition into two-element blocks), order each pair so $i_a<j_a$ and order the pairs by $i_1<\cdots <i_s$; set $\mathrm{sgn}(M)=\mathrm{sgn}(i_1,j_1,\dots ,i_s,j_s)$, the sign of that permutation of $\{1,\dots ,2s\}$. Then $\mathrm{Pf}(A_S)={\sum }_M\mathrm{sgn}(M){\prod }_{a=1}^s{a}_{s_{i_a},s_{j_a}}$. An empty spanning family denotes the zero subspace.