AI 中文总结
研究李代数\(\mathfrak{sl}_n\)在彩色图和多色约翰逊图上的作用，通过将图空间与张量幂等同，用\(\mathfrak{sl}_n\)根算子表示邻接算子并推导谱，还给出了相关应用及结论。
AI 中文摘要
我们考虑在固定的\(N\)个顶点集上的\((n - 1)\)色图空间。完全图\(K_N\)的每条边位置有\(n\)种可能状态，这使得此类图空间与张量幂\((\mathbb{C}^n)^{\otimes m}\)自然等同，其中\(m = \binom{N}{2}\)，并在其上定义了李代数\(\mathfrak{gl}_n\)的对角作用，限制后为\(\mathfrak{sl}_n\)的作用。对于固定轮廓\(\alpha = (\alpha_0,\dots,\alpha_{n - 1})\)，我们考虑图\(J(m;\alpha)\)，其顶点是此轮廓的彩色图，邻接关系由两个边位置状态的单次交换定义。该图是固定轮廓单词集上的换位图，也称为“多层”。主要结果是用\(\mathfrak{sl}_n\)的根算子表示邻接算子，并通过\(\mathfrak{gl}_n\)的二次卡西米尔算子和舒尔 - 外尔分解推导其谱。证明了邻接算子属于代数\(\End_{S_m}(\mathcal{C}_\alpha)\)的中心。用科斯特卡数和斯皮赫特模的维数描述了每个谱块对相应特征值重数的贡献。对于\(n = 2\)，得到经典约翰逊图及其已知谱。作为应用，建立了价公式，证明了连通性，得到独立集的霍夫曼界，并详细考虑了三态情况；在此情况下，自然对称子空间实现了模\(\Sym^m(\mathbb{C}^3)\)。
英文摘要
We consider the space of $(n-1)$-colored graphs on a fixed set of $N$ vertices. Each edge position of the complete graph $K_N$ has $n$ possible states: the absence of an edge and $n-1$ colors. This gives a natural identification of the space of such graphs with the tensor power $(\mathbb C^n)^{\otimes m}$, where $m=\binom N2$, and defines on it the diagonal action of the Lie algebra $\mathfrak{gl}_n$, and, after restriction, the action of $\mathfrak{sl}_n$. For a fixed profile $α=(α_0,\dots,α_{n-1})$, we consider the graph $J(m;α)$ whose vertices are colored graphs of this profile and whose adjacency is defined by a single exchange of states in two edge positions. This graph is the transposition graph on the set of words with fixed profile, also known as the \emph{multislice}. The main result is an expression of the adjacency operator in terms of the root operators of $\mathfrak{sl}_n$ and a derivation of its spectrum by means of the quadratic Casimir operator of $\mathfrak{gl}_n$ and the Schur--Weyl decomposition. It is proved that the adjacency operator belongs to the center of the algebra $\End_{S_m}(\mathcal C_α)$. The contribution of each spectral block to the multiplicity of the corresponding eigenvalue is described in terms of a Kostka number and the dimension of a Specht module. For $n=2$, one obtains the classical Johnson graph and its known spectrum. As applications, a formula for the valency is established, connectivity is proved, the Hoffman bound for independent sets is obtained, and the three-state case is considered in detail; in this case the natural symmetrized subspace realizes the module $\Sym^m(\mathbb C^3)$.
Comments18 pages
