arXiv subjects

Zehavit Zvi

Publications and source records attributed to Zehavit Zvi.

4 recordsLinked to original sources

arXiv2026

Permutation of edges in mutation reduction of pointed Brauer trees

Aihara developed an algorithm for Brauer tree algebras, which we call a mutation reduction, for getting from a Brauer tree algebra to the simpler Brauer star algebra using a sequence of mutations centered on edges. Schaps and Zvi, using the Schaps-Zakay theory of pointing the tree, showed that different algorithms for the sequence of mutations give permutations of the edges. Kozakai gave a new algorithm for a mutation reduction that depends on a given pointing and describes the evolution of the pointing under the mutation reduction. In this paper, we define a pointed generalized Aihara algorithm and show that its permutation is the identity. We give a general form for the permutations resulting from Kozakai's algorithm, which we illustrate with examples from uni-branch binary trees.

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arXiv2026

Efficiency of mutation reduction for Brauer trees

Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. Aihara develped an algorithm, which we call a mutation reduction, for getting from a Brauer tree algebra to the simpler Brauer star algebra using a sequence of mutations centered on edges. Schaps and Zvi, using the Schaps-Zakay theory of pointing the tree, showed that different algorithms for the sequence of mutations give permutations of the edges. We give a modification of Aihara's algorithm and tested it against the original algorithm for computational efficiency. We prove that all versions of Aihara's algorithm are the fastest possible in the sense of requiring the least number steps to reach the Brauer star, when compared to all possible complete mutation reduction algorithms.

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arXiv2024

Elementary equivalences for blocks with normal elementary abelian defect group of rank 2

We consider the effect of performing an elementary equivalence as defined by Okuyama on a group block of form $F(C_p \times C_p)\rtimes C_r$, for a field $F$ of characteristic $p$. If $I=\{0,1,2,\dots r-1\}$ is the set of residues corresponding to the simple modules of $FC_r$, the elementary equivalence is determined by a proper, non-empty subset $I_0 \subset I$, and the corresponding elementary tilting complex is completely determined by $I_0$. We give a catalog of homogeneous maps between irreducible components of the elementary tilting complex. When the subset $I_0$ is an interval, we prove that the maps in the catalog are sufficient to describe all homogeneous maps between two irreducible components.

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arXiv2016

Mutations and Pointing for Brauer Tree Algebras

Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between the ordinary Brauer tree algebra and a particular algebra called the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the star-algebra of the Brauer tree. Our main purpose in this research is to combine these two approaches: We find an algorithm for which we are able to obtain a tilting complex constructed from irreducible complexes of length two {[}SZ1{]}, which is obtained from a sequence of mutations and corresponds to the star-to-tree complex for the pointing given by a reversed Green's walk. For the algorithm given by Aihara in \cite{Ai}, we prove that Aihara's tilting complex can be obtained from the completely folded Rickard tree-to-star complex with left alternating pointing by a permutation of projectives corresponding to the cyclic ordering of edges at vertices of non-zero even distance from the exceptional vertex. The natural numbering of the Aihara algorithm can be optained from the left alternating pinting by the inverse of this permutation.

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