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Mary Schaps

Publications and source records attributed to Mary Schaps.

12 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.

math.CO↗
arXiv2025

Quotients of Buildings by Non-uniform Lattices

We consider quotients of the Bruhat-Tits building associated to the projective linear groups of dimension $d>2$ over the function field $\mathbb F_q(t)$ by a non-uniform lattice $\Gamma$ which is a congruence subgroup in the non-uniform lattice $ PGL_{d}(R)$, where $R=\mathbb F_q[\frac{1}{t}]$. We determine a fundamental domain and demonstrate that the quotient, while not cofinite, is at least of finite covolume. We do the case $d=3$ in considerable detail.

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arXiv2024

Spin Multipartitions

We conjecture an algorithm to construct spin multipartitions and prove that all the level one Fock spaces using our combinatorics are modules over the quantum enveloping algebra.

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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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arXiv2023

Faces in Crystals of Affine Type A and the Shape of Canonical Basis Elements

For a dominant integral weight $\Lambda$ in a Lie algebra of affine type A and rank $e$, and an interval $I_0$ in the residue set $I$, we define the face for the interval $I_0$ to be the subgraph of the block-reduced crystal $\widehat P(\Lambda)$ that is generated by $f_i$ for $i \in I_0$. We show that such a face has an automorphism that preserves defects. For an interval of length $2$, we also give a non-recursive construction of the $e$-regular multipartitions with weights in the face, as well as a formula for the number of $e$-regular multipartitions at each vertex of the face. For an affine Lie algebra of type $A$ we define and investigate the shape of canonical basis elements, a sequence counting the number of multipartitions with a given coefficient. For finite faces generated by $\Lambda$ with $|I_0|=1,2$, we give a non-recursive closed formula for the canonical basis elements.

math.RT↗
arXiv2021

External vertices for crystals of type A

We show that a vertex in the reduced crystal is i-external for a residue i if the defect is less than the absolute value of the i-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a given defect d are external in at least one i-string. Combining this with the Chuang-Rouquier categorification for the simple modules of the cyclotomic Hecke algebras of type A and rank e, this would imply a version of Donovan's Conjecture for the cyclotomics. For e=2, we calculate an approximation to this bound.

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arXiv2020

Nonrecursive canonical basis computations for low rank Kashiwara crystals of type A

For symmetric Kashiwara crystals of type $A$ and rank $e=2$, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a non-recursive manner. In particular, for a symmetric crystal with $Λ=a Λ_0+a Λ_1$, we give formulae for the canonical basis elements for all the $e$-regular multipartitions with defects either $k(a-k)$ or $k(a-k)+2a$, for $0 \leq k \leq a$.

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arXiv2020

External Littelmann paths for crystals of type A

For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight $Λ$, there is a labeling by multipartitions and by piecewise linear paths in the real weight space called Littelmann paths. Both labelings are constructed recursively, but since Kashiwara demonstrated that the crystals are isomorphic, there is a bijection between the labels. We choose a multicharge $(k_1,\dots,k_r)$, with $0 \leq k_1\leq k_2....\leq k_r \leq e-1$. We put $k_i$ in the node at the upper left corner of partition $i$ of the multipartition and let the residues from $\mathbb Z/ e \mathbb Z$ increase across rows and decrease down columns. For e=2, we call a multipartition residue-homogeneous if all nonzero rows end in nodes of the same residue and partitions with the same corner residue have first rows of the same parity. It is strongly residue homogeneous if each partition ends in a triangle of whose side has length one less than the first row of the next partition. In this paper we show that each such multipartition corresponds to a Littelmann path which is unidirectional in the sense that the projection of the the main part of the path to the coordinates of the fundamental weights consists of long paths all lying in either the second or fourth quadrant, separated by oscillating paths with a fixed integer oscillator. The path corresponding to such a multipartition can be constructed non-recursively using only integers describing the structure of the multipartition.

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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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arXiv2010

sl_2-actions along short strings for spin blocks

The problem of source algebra equivalences between blocks at the ends of the maximal strings of spin blocks of the symmetric and alternating groups has recently been settled, but so far there has not even been a candidate of the tilting complex defining the reflections of the internal blocks of the string. Our aim in this paper is to propose a definition for the mappings E_i and F_i and for their divided powers. The solution we propose here is to halve the operators only when both halves are isomorphic, which, on the level of the Grothendieck group, corresponds to working with paired simples as pairs, and crossing over between the symmetric and alternating groups, in order to send paired simples to paired simples.This is equivalent to working with the irreducible supermodules, as in the work of Brundan and Kleshchev, though we don't introduce the supermodule language into this paper. The main difference between our approach and theirs is the crossovers, which occasionally force extra halving of modules.The definition is based on the method of permutation m odules from Kessar-Schaps. We apply it to short strings and show that, at least on the individual strings, it is compatible with the form of the tilting complex used by Chuang and Rouquier.

math.RT↗
arXiv2007

Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3-bound syspi_1(X_{\PCH}) > 4/3 \log(g(X_{\PCH})). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

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