论文标题
通过量化的Clifford代表
Skew Howe duality for $U_q(\mathfrak{gl}_n)$ via quantized Clifford algebras
论文作者
论文摘要
我们通过使用量化的Clifford代数中的双中心属性在量化的Clifford代数中使用双重centralizer属性，开发了通用线性量子组$ u_q（\ Mathfrak {gl} _n）$不变理论的第一个基本定理的操作员。特别是，我们表明$ u_q（\ mathfrak {gl} _m）$生成了$ u_q（\ mathfrak {gl} _n）$ - 对编织外部代数的张量产物的概括的键盘（\ mathfrak {gl} _n）$ - 我们获得了$ u_q（\ Mathfrak {gl} _n）\ otimes u_q（\ Mathfrak {gl} _M）$ - module $ \ bigWedge_q（\ mathbb {c}^n）^n） \ bigWedge_q（\ Mathbb {C}^{nm}）$通过计算显式关节最高权重向量。我们发现，这种分解中的不可值得模块的参数为与众所周知的偏斜$ gl_n \ times gl_m $ duality相同的主要权重。 Clifford代数是我们工作的重要特征：它们为经典和量化的偏差二元性结果提供了一个统一的框架，可以扩展到包括类型的$ \ Mathbf {bd} $的正交代数。
We develop an operator commutant version of the First Fundamental Theorem of invariant theory for the general linear quantum group $U_q(\mathfrak{gl}_n)$ by using a double centralizer property inside a quantized Clifford algebra. In particular, we show that $U_q(\mathfrak{gl}_m)$ generates the centralizer of the $U_q(\mathfrak{gl}_n)$-action on the tensor product of braided exterior algebras $\bigwedge_q(\mathbb{C}^n)^{\otimes m}$. We obtain a multiplicity-free decomposition of the $U_q(\mathfrak{gl}_n) \otimes U_q(\mathfrak{gl}_m)$-module $\bigwedge_q(\mathbb{C}^n)^{\otimes m} \cong \bigwedge_q(\mathbb{C}^{nm})$ by computing explicit joint highest weight vectors. We find that the irreducible modules in this decomposition are parametrized by the same dominant weights as in the classical case of the well-known skew $GL_n \times GL_m$-duality. Clifford algebras are an essential feature of our work: they provide a unifying framework for classical and quantized skew Howe duality results that can be extended to include orthogonal algebras of types $\mathbf{BD}$.
