论文标题
在转移的利特伍德 - 里查森系数和利特伍德 - 雷克森的系数上
On the Shifted Littlewood-Richardson Coefficients and Littlewood-Richardson Coefficients
论文作者
论文摘要
我们对移动的Littlewood-Richardson系数$ f_ {λμ}^ν$（$λ，μ，μ，μ，是严格的分区）的新解释。系数$ g_ {λμ} $出现在schur $ q $ - function $q_λ$的分解中的$q_λ$中的总和$q_λ= 2^{l（λ）} \sum_μg_{λμg_{λμ}s_μ$，可以被视为$ f_ iS $ f_ iS $ f_ iS $ lentur $ L（λ）$）。我们还将$ g_ {λμ} $的另一个描述作为集合的子集的基数，该集合计数Littlewood-Richardson系数$ C_ {μ^Tμ}^{\tildeλ} $。这种新的观点使我们能够在$ g_ {λμ} $和$ c_ {μ^tμ}^{\tildeλ} $之间建立连接。更准确地说，我们证明$ g_ {λμ} = g_ {λμ^t} $，$ g_ {λμ} \ leq c_ {μ^tμ}^{\tildeλ} $。我们猜想$ g_ {λμ}^2 \ leq c^{\tildeλ} _ {μ^tμ} $，并在我们的组合模型上提出一些猜想，这意味着如果有效，这将暗示这种不平等。
We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{λμ}^ν$ ($λ,μ,ν$ are strict partitions). The coefficients $g_{λμ}$ which appear in the decomposition of Schur $Q$-function $Q_λ$ into the sum of Schur functions $Q_λ= 2^{l(λ)}\sum_μg_{λμ}s_μ$ can be considered as a special case of $f_{λμ}^ν$ (here $λ$ is a strict partition of length $l(λ)$). We also give another description for $g_{λμ}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{μ^tμ}^{\tildeλ}$. This new point of view allows us to establish connections between $g_{λμ}$ and $c_{μ^t μ}^{\tildeλ}$. More precisely, we prove that $g_{λμ}=g_{λμ^t}$, and $g_{λμ} \leq c_{μ^tμ}^{\tildeλ}$. We conjecture that $g_{λμ}^2 \leq c^{\tildeλ}_{μ^tμ}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.
