Irreducible representations of the braid group B3B_3 in dimension 6

We use $q$-Pascal's triangle to define a family of representations of dimension 6 of the braid group $B_3$ on three strings. Then we give a necessary and sufficient condition for these representations to be irreducible

B3B_3 block representations of dimension 6 and braid reversions

We construct a family of six dimensional block representations of the braid group $B_3$ on three strings. We show that these representations can be used to distinguish braids of knots with 9 and 10 crossings from their reversed braids

Mechanisms of activation of innate-like intraepithelial T lymphocytes

Intraepithelial T lymphocytes (T-IEL) contain subsets of innate-like T cells that evoke innate and adaptive immune responses to provide rapid protection at epithelial barrier sites. In the intestine, T-IEL express variable T cell antigen receptors (TCR), with unknown antigen specificities. Intriguingly, they also express multiple inhibitory receptors, many of which are normally found on exhausted or antigen-experienced T cells. This pattern suggests that T-IEL are antigen-experienced, yet it is not clear where, and in what context, T-IEL encounter TCR ligands. We review recent evidence indicating TCR antigens for intestinal innate-like T-IEL are found on thymic or intestinal epithelium, driving agonist selection of T-IEL. We explore the contributions of the TCR and various co-stimulatory and co-inhibitory receptors in activating T-IEL effector functions. The balance between inhibitory and activating signals may be key to keeping these highly cytotoxic, rapidly activated cells in check, and key to harnessing their immune surveillance potential

Chronic inflammation permanently reshapes tissue-resident immunity in celiac disease

Tissue-resident lymphocytes play a key role in immune surveillance, but it remains unclear how these inherently stable cell populations respond to chronic inflammation. In the setting of celiac disease (CeD), where exposure to dietary antigen can be controlled, gluten-induced inflammation triggered a profound depletion of naturally occurring Vγ4+/Vδ1+ intraepithelial lymphocytes (IELs) with innate cytolytic properties and specificity for the butyrophilin-like (BTNL) molecules BTNL3/BTNL8. Creation of a new niche with reduced expression of BTNL8 and loss of Vγ4+/Vδ1+ IELs was accompanied by the expansion of gluten-sensitive, interferon-γ-producing Vδ1+ IELs bearing T cell receptors (TCRs) with a shared non-germline-encoded motif that failed to recognize BTNL3/BTNL8. Exclusion of dietary gluten restored BTNL8 expression but was insufficient to reconstitute the physiological Vγ4+/Vδ1+ subset among TCRγδ+ IELs. Collectively, these data show that chronic inflammation permanently reconfigures the tissue-resident TCRγδ+ IEL compartment in CeD

Interleukin-15 promotes intestinal dysbiosis with butyrate deficiency associated with increased susceptibility to colitis

Dysbiosis resulting in gut-microbiome alterations with reduced butyrate production are thought to disrupt intestinal immune homeostasis and promote complex immune disorders. However, whether and how dysbiosis develops before the onset of overt pathology remains poorly defined. Interleukin-15 (IL-15) is upregulated in distressed tissue and its overexpression is thought to predispose susceptible individuals to and have a role in the pathogenesis of celiac disease and inflammatory bowel disease (IBD). Although the immunological roles of IL-15 have been largely studied, its potential impact on the microbiota remains unexplored. Analysis of 16S ribosomal RNA-based inventories of bacterial communities in mice overexpressing IL-15 in the intestinal epithelium (villin-IL-15 transgenic (v-IL-15tg) mice) shows distinct changes in the composition of the intestinal bacteria. Although some alterations are specific to individual intestinal compartments, others are found across the ileum, cecum and feces. In particular, IL-15 overexpression restructures the composition of the microbiota with a decrease in butyrate-producing bacteria that is associated with a reduction in luminal butyrate levels across all intestinal compartments. Fecal microbiota transplant experiments of wild-type and v-IL-15tg microbiota into germ-free mice further indicate that diminishing butyrate concentration observed in the intestinal lumen of v-IL-15tg mice is the result of intrinsic alterations in the microbiota induced by IL-15. This reconfiguration of the microbiota is associated with increased susceptibility to dextran sodium sulfate-induced colitis. Altogether, this study reveals that IL-15 impacts butyrate-producing bacteria and lowers butyrate levels in the absence of overt pathology, which represent events that precede and promote intestinal inflammatory diseases

Dietary Gluten Permanently Reshapes the Tissue-Resident Intraepithelial T Cell Compartment in Celiac Disease

Tissue-resident lymphocytes play a key role in immune surveillance, but it remains unclear how these inherently stable cell populations respond to chronic inflammation. In the setting of celiac disease (CeD), where exposure to dietary antigen can be controlled, gluten-induced inflammation triggered a profound depletion of naturally-occurring V4⁺V1⁺ intraepithelial lymphocytes (IELs) with innate cytolytic properties and specificity for the butyrophilin-like (BTNL) molecules BTNL3/BTNL8. Creation of a new niche with reduced expression of BTNL8 and loss of V4⁺V1⁺ IELs was accompanied by the expansion of gluten-sensitive, interferon--producing V1⁺ IELs bearing T cell receptors (TCRs) with a shared non-germline-encoded motif that failed to recognize BTNL3/BTNL8. Exclusion of dietary gluten restored BTNL8 expression but failed to reconstitute the V4⁺V1⁺ ‘gut signature’ among TCR⁺ IELs. Collectively, these data demonstrate that chronic inflammation permanently reconfigures the tissue-resident TCR⁺ IEL compartment, thus the paradigm of tissue-resident lymphocyte stability needs revisiting

Representations of the necklace braid group NBn{{\mathcal {N}}{\mathcal {B}}}_n of dimension 4 (n=2,3,4n=2,3,4)

AbstractWe consider the irreducible representations each of dimension 2 of the necklace braid group $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ). We then consider the tensor product of the representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) of degree 2 are unitary relative to a hermitian positive definite matrix.</jats:p