论文标题
晶体不变理论II：伪用力
Crystal invariant theory II: Pseudo-energies
论文作者
论文摘要
几何晶体运算符和几何$ r $ $ - matrices（或几何Weyl ofers Action）在$ MN $变量中对合理功能领域进行通勤动作。我们研究了这些动作的各种组合的不变性，我们将其视为$ s_m $，$ {\ rm sl} _m $，$ s_n \ times s_m $，$ {\ rm sl} _n} _n _n _n \ times \，s_m $ $ slm的“晶体类似物” sl} _m $在变量的$ m \ times n $矩阵中作用于多项式环。 $ {\ rm gl} _m $ - 几何$ r $ -matrices生成的$ s_m $ - $ $ - $ - $ $ $ $ $ $ $ $ $ $ $ $ $ $ r $ - matrices的多项式不变性。在作者的先前论文中，$ {\ rm gl} _M $ - 几何晶体运算符的多项式不变性被描述为环环对称函数的子。 在本文中，我们在其余情况下为理性不变的场提供了猜想的生成集，并给出了以这些猜想的发生器来表达大量循环对称函数的公式。我们的结果包括单行几何晶体产物的中央电荷和能量功能的新的正配方，以及Kirillov和Berenstein的新衍生物用于Coarchge。公式表现出这些功能所具有的对称性。
The geometric crystal operators and geometric $R$-matrices (or geometric Weyl group actions) give commuting actions on the field of rational functions in $mn$ variables. We study the invariants of various combinations of these actions, which we view as "crystal analogues" of the invariants of $S_m$, ${\rm SL}_m$, $S_n \times S_m$, ${\rm SL}_n \times \, S_m$, and ${\rm SL}_n \times {\rm SL}_m$ acting on the polynomial ring in an $m \times n$ matrix of variables. The polynomial invariants of the $S_m$-action generated by the ${\rm GL}_m$-geometric $R$-matrices were described by Lam and the third-named author as the ring of loop symmetric functions. In a previous paper of the authors, the polynomial invariants of the ${\rm GL}_m$-geometric crystal operators were described as a subring of the ring of loop symmetric functions. In this paper, we give conjectural generating sets for the fields of rational invariants in the remaining cases, and we give formulas expressing a large class of loop symmetric functions in terms of these conjectural generators. Our results include new positive formulas for the central charge and energy function of a product of single-row geometric crystals, and a new derivation of Kirillov and Berenstein's piecewise-linear formula for cocharge. The formulas manifest the symmetries possessed by these functions.
