某些群代数中群行列式的因子分解
Factorization of group determinant in some group algebras
摘要：
给出了有限阿贝尔群的群代数上的群行列式的因式分解的Frobenius定理的一个类似定理，并将其推广到二面体群和广义四元数群.进一步，我们将二面体群和广义四元数群的群行列式描述为齐次多项式的循环行列式。这个类比比Frobenius定理更强，作为推论，我们得到了群代数中逆元的一个简单表达式。此外，对应于一次表示的群代数上的群行列式的因式分解的不可约因子的分解子具有有趣的代数性质。从这个结果，我们知道一次表示形成自然配对。Frobeinus定理的推广在现阶段还没有表现为行列式。我们期望找到一个类似于Frobenius定理的行列式表达式。
英文摘要：
We give an analog of Frobenius' theorem about the factorization of the group determinant on the group algebra of finite abelian groups and we extend it into dihedral groups and generalized quaternion groups. Furthermore, we describe the group determinant of dihedral groups and generalized quaternion groups as a circulant determinant of homogeneous polynomials. This analog on the group algebra is stronger than Frobenius's theorem and as a corollary, we obtain a simple expression formula for inverse elements in the group algebra. Furthermore, the commutators of irreducible factors of the factorization of the group determinant on the group algebra corresponding to degree one representations have interesting algebraic properties. From this result, we know that degree one representations form natural pairing. At the current stage, the extension of Frobeinus' theorem is not represent as a determinant. We expect to find a determinant expression similar to Frobenius' theorem.
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