arXiv subjects
Publications and source records attributed to Taro Sakurai.
Let $π$ be a set of primes and let $G$ be a finite $π$-separable group. We prove that the Cartan matrix $C$ for the $π$-partial characters of $G$ is equivalent over the integers to a matrix $\operatorname{diag}(|\mathbf{C}_G(x)|_{π'})$, where $x$ runs over a set of representatives of the $π$-classes of $G$. In particular, we prove that $\det C = \prod |\mathbf{C}_G(x)|_{π'}.$
We introduce a procedure based on computational algebraic geometry to determine whether two algebras are isomorphic. We then apply it to show that if $R$ is a commutative unital ring in which $2$ is not invertible, $G$ is a group of order dividing $64$ and $H$ some group, then an isomorphism of unital algebras $RG \cong RH$ implies an isomorphism of groups $G \cong H$.
The Modular Isomorphism Problem asks, if an isomorphism between modular group algebras of finite $p$-groups over a field $F$ implies an isomorphism of the group bases. We explore the differences of knowledge on the problem when $F$ is either assumed to be a prime field or a general field of characteristic $p$. After revising the literature and explaining reasons for the differences, we generalize some of the positive answers to the problem from the prime field case to the general case.
Counterexamples to the Modular Isomorphism Problem were discovered recently. These are non-isomorphic finite $2$-groups $G$ and $H$ that have isomorphic group algebras over the field $\mathbb{Z}/2\mathbb{Z}$ and non-isomorphic group algebras over the $2$-adic integers $\mathbb{Z}_2$. We show that the groups $G$ and $H$ already have non-isomorphic group algebras over the ring $\mathbb{Z}/4\mathbb{Z}$.
We prove that any tensor product factorization with a commutative factor of a modular group algebra over a prime field comes from a direct product decomposition of the group basis. This extends previous work by Carlson and Kovács for the commutative case and answers a question of them in some cases.
The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by García-Lucas, Margolis and del Río [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these negative solutions in the class of two-generated finite $2$-groups with dihedral central quotient, and solve the original question for all groups within this class. As a result, we discover new negative solutions and simple algebra isomorphisms. At the same time, the positive solutions for most of the groups in this class give some insights what makes the negative solutions special.
We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the modular group algebra of a finite $p$-group. Finally, we apply our results to new classes of groups.
We show that the expected number of cliques in the Erdős-Rényi random graph $G(n,p)$ is $n^{\frac1{-2\log p}(\log n-2\log\log n+O(1))}$.
In formal concept analysis, it is well-known that the number of formal concepts can be exponential in the worst case. To analyze the average case, we introduce a probabilistic model for random formal contexts and prove that the average number of formal concepts has a superpolynomial asymptotic lower bound.
Finite groups with very few character values are characterized. The following is the main result of this article: a finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a non-trivial elementary abelian $3$-group. The proof involves the analysis of the centralizers of involutions.
For a prime $p$, we determine a Sylow $p$-subgroup $D$ of a finite group $G$ such that the principal $p$-block $B$ of $G$ has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov thirty years ago. Our proof relies on the classification of finite simple groups.
Let $R$ be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over $R$. Our main result states that if $G$ is a hereditary group over $R$ then a unital algebra isomorphism between group algebras $RG \cong RH$ implies a group isomorphism $G \cong H$ for every finite group $H$. As application, we study the modular isomorphism problem, which is the isomorphism problem for finite $p$-groups over $R = \mathbb{F}_p$ where $\mathbb{F}_p$ is the field of $p$ elements. We prove that a finite $p$-group $G$ is a hereditary group over $\mathbb{F}_p$ provided $G$ is abelian, $G$ is of class two and exponent $p$ or $G$ is of class two and exponent four. These yield new proofs for the theorems by Deskins and Passi-Sehgal.
An explicit formula for a weight enumerator of linear-congruence codes is provided. This extends the work of Bibak and Milenkovic [IEEE ISIT (2018) 431-435] addressing the binary case to the non-binary case. Furthermore, the extension simplifies their proof and provides a complete solution to a problem posed by them.
In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block $B$ with $k(B) = 1$ where $k(B)$ is the number of irreducible ordinary characters of $B$. In 1982, Brandt established a characterization of a block with defect group of order two as a block $B$ with $k(B) = 2$. These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively. In this paper, we redefine $k(A)$ to be the codimension of the commutator subspace $K(A)$ of a finite-dimensional algebra $A$ and prove analogous statements for arbitrary (not necessarily symmetric) finite-dimensional algebras. This is achieved by extending the Okuyama refinement of the Brandt result to this setting. To this end, we study the codimension of the sum of the commutator subspace $K(A)$ and $n$th Jacobson radical $\operatorname{Rad}^n(A)$. We prove that this is Morita invariant and give an upper bound for the codimension as well.
From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th (right) socle $ZS^n(A) := Z(A) \cap \operatorname{Soc}^n(A)$ of a finite-dimensional algebra $A$ is a Morita invariant; This is a generalization of important Morita invariants --- the center $Z(A)$ and the Reynolds ideal $ZS^1(A)$. As an example, we also studied $ZS^n(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent set of $ZS^n(FG)$. In fact, such elements form a basis of $ZS^n(FG)$ for every integer $1 \le n \le p$ if $G$ is powerful. As a corollary we have $\operatorname{Soc}^p(FG) \subseteq Z(FG)$ if $G$ is powerful.
Slicing a module into semisimple ones is useful to study modules. Loewy structures provide a means of doing so. To establish the Loewy structures of projective modules over a finite dimensional symmetric algebra over a field $F$, the Landrock lemma is a primary tool. The lemma and its corollary relate radical layers of projective indecomposable modules to radical layers of the $F$-duals of those modules ("dual symmetry") and to socle layers of those modules ("reciprocity"). We generalize these results to an arbitrary finite dimensional algebra $A$. Our main theorem, which is the same as the Landrock lemma for finite dimensional symmetric algebras, relates radical layers of projective indecomposable modules $P$ to radical layers of the $A$-duals of those modules and to socle layers of injective indecomposable modules $νP$ where $ν({-})$ is the Nakayama functor. A key tool to prove the main theorem is a pair of adjoint functors, which we call socle functors and capital functors.