AI 中文总结
该研究基于Kim–Rhoades的工作，给出B型费米子不变量环的双分次Frobenius级数显式公式，确定相关特征标重数，首次得到A、B型中对所有参数成立的非平凡特征标结果。
AI 中文摘要
设$\u27E8$表示超八面体群。B型不变量环$R_{\mathfrak{B}_n}^{(k,j)}$是由$k$组$n$个交换变量和$j$组$n$个反交换变量构成的多项式环，模去由无常数项的对角$\u27E8$-不变量生成的理想所得的商环。基于Kim–Rhoades（2022）的工作，我们给出了$R_{\mathfrak{B}_n}^{(0,2)}$的双分次Frobenius级数的显式公式：每个不可约$\u27E8$-特征标的双分次重数是单个Schur多项式，因此$R_{\mathfrak{B}_n}^{(0,2)}$作为$\u27E8_2 \times \u27E8$-模是无重数的。随后我们确定，$R_{\mathfrak{B}_n}^{(0,3)}$的符号特征标的三分次重数由单个Schur函数给出。最后，对所有$k$和$j$，我们确定了A型不变量环$R_{n}^{(k,j)}$中标准特征标的重数，以及$R_{\mathfrak{B}_n}^{(k,j)}$中由双分划$((n-1),(1))$和$((n-1,1),\varnothing)$标记的特征标的重数。这些是首次在A型或B型中对所有$(k,j)$确立的非平凡特征标结果。
英文摘要
Let $\mathfrak{B}_n$ denote the hyperoctahedral group. The type $B$ coinvariant rings $R_{\mathfrak{B}_n}^{(k,j)}$ are quotients of the ring of polynomials in $k$ sets of $n$ commuting variables and $j$ sets of $n$ anticommuting variables by the ideal generated by the diagonal $\mathfrak{B}_n$-invariants without constant term. Building upon the work of Kim--Rhoades (2022), we give an explicit formula for the bigraded Frobenius series of $R_{\mathfrak{B}_n}^{(0,2)}$: the bigraded multiplicity of each irreducible $\mathfrak{B}_n$-character is a single Schur polynomial, so $R_{\mathfrak{B}_n}^{(0,2)}$ is multiplicity-free as a $\operatorname{GL}_2 \times \mathfrak{B}_n$-module. We then determine that the trigraded multiplicity of the sign character of $R_{\mathfrak{B}_n}^{(0,3)}$ is given by a single Schur function. Finally, for all $k$ and $j$, we determine the multiplicity of the standard character in the type $A$ coinvariant ring $R_{n}^{(k,j)}$, as well as the multiplicities of the characters indexed by the bipartitions $((n-1),(1))$ and $((n-1,1),\varnothing)$ in $R_{\mathfrak{B}_n}^{(k,j)}$. These are the first nontrivial characters established for all $(k,j)$ in either of types $A$ or $B$.
Comments34 pages, 4 tables, 1 figure. Comments are welcome
