论文标题
投影连接光谱和$ \ tilde {a} _n $的表示形式
Projective Joint Spectra and Characters of representations of $\tilde{A}_n$
论文作者
论文摘要
对于正方形复杂值$ n \ times n $矩阵$ a_1，\ dots，a_n $的元组，其线性组合的决定因素$ x_1a_1 +\ cdots +x_na_n $，称为\ textit {a p-colien}，称为$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ c [x n $ n $ n $ c [x n $ n $ c [x]该多项式的零集是射影空间中的代数集$ \ c \ po^{n-1} $。该集合称为元组$（a_1，...，a_n）$的确定性超曲面或确定歧管。在杜鹃，斯特森，特切尔夫（2021）中显示，如果$ g $是一个非特殊的$ a，b $ a，b $或$ d $，$ρ_1$和$ρ_2$的类型的coxer组，则是$ g $的两个线性表示，是$ g $的两个线性表示，$ g $的$ g $ $ g $ $ g $ $ quct $ quct $ quct $ quct_ $ quct $ quct $ quct_ $ c $ cox $ cox_1 $ quct_1射影空间中的除数，$ρ_1$和$ρ_2$的字符相等，因此，$ρ_1$和$ρ_2$等效。在Peebles中，Tchernev Stestesin（准备）该结果扩展到了Aggine Coxeter组的角色一部分，包括$ b，c $和$ d $。在那里表明，每个这样的组都包含一个有限的子集，因此，如果该集合在两个有限维表示下的确定性超出图像作为投影空间中的除数重合，则这些表示的字符相等。值得注意的是，此结果不涵盖$ a $类型的仿型coxeter组，因为它们的组合材料完全不同。本文我们明确地在$ \ tilde {a} _n $中明确构建了一个有限的设置，具有相同的属性。我们还表明，每个群体是细分组的半领产品和有限生成的Abelian组都包含具有相似属性的有限子集：对于该组的每个有限二维表示，集合图像的确定性超出表情都确定表示字符。
For a tuple of square complex-valued $N\times N$ matrices $A_1,\dots,A_n$ the determinant of their linear combination $x_1A_1+\cdots +x_nA_n$, which is called \textit{a pencil}, is a homogeneous polynomial of degree $N$ in $\C[x_1,...x_n]$. Zero-set of this polynomial is an algebraic set in the projective space $\C\Po^{n-1}$. This set is called the determinantal hypersurface or determinantal manifold of the tuple $(A_1,...,A_n)$. It was shown in Cuckovic, Stessin, Tchernev (2021) that if $G$ is a non-special Coxeter group of type $A,B$, or $D$, $ρ_1$ and $ρ_2$ are two linear representations of $G$, and the determinantal hypersurfaces of images of the Coxeter generators of $G$ under $ρ_1$ and $ρ_2$ coincide as divisors in the projective space, the characters of $ρ_1$ and $ρ_2$ are equal, and, therefore, $ρ_1$ and $ρ_2$ are equivalent. In Peebles, Stessin, Tchernev (in preparation) this result was extended in the characters part to affine Coxeter groups of types $B,C$, and $D$. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of $A$ type are not covered by this result, as their combinatorics is quite different.mIn this paper we explicitly construct a finite set in $\tilde{A}_n$ having the same property. We also show that every group which is a semidirect product of a fine group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensonal representation of the group, the determinantal hypersurface of images of the set determines the representation character.
