论文标题
小小的林木 - 里查森系数的竞标
Bidilatation of Small Littlewood-Richardson Coefficients
论文作者
论文摘要
Littlewood-Richardson系数$ c^ν_{λ，μ} $是张量产品分解中的多重性，该分解是通用线性组GL $（n，{\ Mathbb c}）$的两个不可减至的表示。它们由最多$ n $的分区$（λ，μ，ν）$的分区$（λ，μ，ν）$进行参数。由所谓的富尔顿猜想，如果$ c^ν_{λ，μ} = 1 $，则$ c^{kν} _ {kλ，kμ} = 1 $，对于任何$ k \ geq 0 $。同样，如Ikenmeyer或Sherman所证明的，如果$ c^ν_{λ，μ} = 2 $，则$ c^{kν} _ {kλ，kμ} = k + 1 $，对于任何$ k \ geq 0 $。在这里，给定一个分区$λ$，我们设置了$λ（p，q）= p（qλ'）'$，其中prime表示共轭分区。我们观察到富尔顿的猜想意味着，如果$ c^ν_{λ，μ} = 1 $，则$ c^{ν（p，q）} _ {λ（p，q），μ（p，q）} = 1 $，对于任何$ p，q q \ geq 0 $。我们的主要结果是，如果$ c^ν_{λ，μ} = 2 $然后$ c^{ν（p，q）} _ {λ（p，q），μ（p，q）} $是二项式$ \ begin {pmatrix {pmatrix} p+q \ q \\ q \\ q \\ q \\ q \\ q \\ q \\ q \ \ q \\ ppmatrix $ $ $ $ $
The Littlewood-Richardson coefficients $c^ν_{λ,μ}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the triples of partitions $(λ, μ, ν)$ of length at most $n$. By the so-called Fulton conjecture, if $c^ν_{λ,μ}=1$ then $c^{kν}_{kλ,kμ}= 1$, for any $k \geq 0$. Similarly, as proved by Ikenmeyer or Sherman, if $c^ν_{λ,μ}=2$ then $c^{kν}_{kλ,kμ} = k + 1$, for any $k\geq 0$. Here, given a partition $λ$, we set $λ(p, q) = p(qλ')'$ , where prime denotes the conjugate partition. We observe that Fulton's conjecture implies that if $c^ν_{λ,μ}=1$ then $c^{ν(p,q)}_{λ(p,q),μ(p,q)}=1$, for any $p, q \geq 0$. Our main result is that if $c^ν_{λ,μ}=2$ then $c^{ν(p,q)}_{λ(p,q),μ(p,q)}$ is the binomial $\begin{pmatrix} p+q\\ q \end{pmatrix}$, for any $p, q \geq 0$.
