论文标题
偏斜双重性和Q-Krawtchouk多项式合奏
Skew Howe duality and q-Krawtchouk polynomial ensemble
论文作者
论文摘要
我们考虑将外部代数$ \ bigWedge \ left（\ mathbb {c}^{n} {n} \ otimes \ left（\ mathbb {c}^{c}^{k} {k} \ right）^{*} \ right）$ a as a $ gl_ glties n time as a y time，不可约合$ gl_ {n} \ times gl_ {k} $表示由成对的年轻图（λ，\barλ'）$参数化，其中$ \barλ'$是$ n \ times k $ retctangle中的补充偶联图为$λ$。我们将图的概率设置为对相应不可约组件的字符的归一化专业化。对于主体专业化，我们得到的概率等于外代代数的$ q $ dimension的$ q $数的比率。我们证明了这种概率分布可以通过Q-Krawtchouk多项式集合来描述。我们得出极限形状，并证明当$ n，k $倾向于无穷大，$ q $以可比的速度趋向于一个时，在极限中的中心限制定理。
We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by pairs of Young diagrams $(λ,\barλ')$, where $\barλ'$ is the complement conjugate diagram to $λ$ inside the $n\times k$ rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the $q$-dimension for the irreducible component over the $q$-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when $n,k$ tend to infinity and $q$ tends to one at comparable rates.
