A Combinatorial Formula for Orthogonal Idempotents in the 00-Hecke Algebra of the Symmetric Group

Building on the work of P.N. Norton, we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the $0$-Hecke algebra of the symmetric group, $\mathbb{C}H_0(S_N)$. This construction is compatible with the branching from $S_{N-1}$ to $S_{N}$.Comment: 25 pages, 2 figure

Excursions into Algebra and Combinatorics at q=0q=0

We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of \JJ-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\tilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thi\'ery on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simply-laced crystal bases.Comment: 92 pages, 13 figures. PHd Dissertation accepted at the University of California on July 15th, 2011. arXiv admin note: text overlap with arXiv:1012.136

On the representation theory of finite J-trivial monoids

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both from a theoretical and computational point of view, and with an implementation in Sage. Motivated by ongoing work on related monoids associated to Coxeter systems, and building on well-known results in the semi-group community (such as the description of the simple modules or the radical), we describe how most of the data associated to the representation theory (Cartan matrix, quiver) of the algebra of any J-trivial monoid M can be expressed combinatorially by counting appropriate elements in M itself. As a consequence, this data does not depend on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M| and m is the number of generators. Along the way, we construct a triangular decomposition of the identity into orthogonal idempotents, using the usual M\"obius inversion formula in the semi-simple quotient (a lattice), followed by an algorithmic lifting step. Applying our results to the 0-Hecke algebra (in all finite types), we recover previously known results and additionally provide an explicit labeling of the edges of the quiver. We further explore special classes of J-trivial monoids, and in particular monoids of order preserving regressive functions on a poset, generalizing known results on the monoids of nondecreasing parking functions.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated comments by referee in version

In Search for a Generalizable Method for Source Free Domain Adaptation

Source-free domain adaptation (SFDA) is compelling because it allows adapting an off-the-shelf model to a new domain using only unlabelled data. In this work, we apply existing SFDA techniques to a challenging set of naturally-occurring distribution shifts in bioacoustics, which are very different from the ones commonly studied in computer vision. We find existing methods perform differently relative to each other than observed in vision benchmarks, and sometimes perform worse than no adaptation at all. We propose a new simple method which outperforms the existing methods on our new shifts while exhibiting strong performance on a range of vision datasets. Our findings suggest that existing SFDA methods are not as generalizable as previously thought and that considering diverse modalities can be a useful avenue for designing more robust models

Identification of genomic groups in the genus Stenotrophomonas using gyrB RFLP analysis

Stenotrophomonas maltophilia isolates have been recovered from various clinical samples, including the respiratory tract of cystic fibrosis (CF) patients, but this organism is also widespread in nature. Previously it has been shown that there is a considerable genomic diversity within S. maltophilia . The aims of our study were to determine the taxonomic resolution of restriction fragment length polymorphism (RFLP) analysis of the polymerase chain reaction-amplified gyrB gene for the genus Stenotrophomonas . Subsequently, we wanted to use this technique to screen a set of S. maltophilia isolates (with emphasis on a specific subset, isolates recovered from CF patients), to assess the genomic diversity within this group. In this study we investigated 191 Stenotrophomonas sp. isolates (including 40 isolates recovered from CF patients) by means of gyrB RFLP. The taxonomic resolution of gyrB RFLP, and hence its potential as an identification tool, was confirmed by comparison with results from published and novel DNA–DNA hybridisation experiments. Our data also indicate that the majority of CF isolates grouped in two clusters. This may indicate that isolates from specific genomic groups have an increased potential for colonisation of the respiratory tract of CF patients.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72378/1/S0928-8244_03_00307-9.pd