scholarly journals Crystal Structures for Double Stanley Symmetric Functions

10.37236/8872 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Graham Hawkes
Keyword(s):  
Crystal Structures ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Type A ◽  
Combinatorial Objects ◽  
Type C ◽  
Stanley Symmetric Functions ◽  
Relationship Of

We relate the combinatorial definitions of the type $A_n$ and type $C_n$ Stanley symmetric functions, via a combinatorially defined "double Stanley symmetric function," which gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives the type $C$ case at $(\mathbf{x},\mathbf{x})$.  We induce a  type $A$ bicrystal structure on the underlying combinatorial objects of this function which has previously been done in the type $A$ and type $C$ cases.  Next we prove a few statements about the algebraic relationship of these three Stanley symmetric functions. We conclude with some conjectures about what happens when we generalize our constructions to type $C$.

scholarly journals Crystal Analysis of type C Stanley Symmetric Functions

10.37236/6952 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Graham Hawkes ◽  
Kirill Paramonov ◽  
Anne Schilling
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Combinatorial Description ◽  
Highest Weight ◽  
Type C ◽  
Stanley Symmetric Functions

Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.

scholarly journals A $t$-generalization for Schubert Representatives of the Affine Grassmannian

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Avinash J. Dalal ◽  
Jennifer Morse
Keyword(s):  
Weyl Group ◽  
Transition Matrix ◽  
Symmetric Functions ◽  
Type A ◽  
Affine Grassmannian ◽  
Combinatorial Objects ◽  
Affine Weyl Group ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Extra Parameter

International audience We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms. Nous présentons deux familles de fonctions symétriques dépendant d'un paramètre $t$ et dont les spécialisations à $t=1$ correspondent aux classes de Schubert dans la cohomologie et l'homologie des variétés Grassmanniennes affines. Les familles sont définies par des statistiques sur certains objets combinatoires associés au groupe de Weyl affine de type $A$ et leurs matrices de transition dans la base des polynômes de Hall-Littlewood sont $t$-positives. Nous conjecturons qu'une de ces familles correspond aux $k$-atomes.

scholarly journals Schur $P$-positivity and Involution Stanley Symmetric Functions

2017 ◽  
Vol 2019 (17) ◽  
pp. 5389-5440 ◽  
Author(s):  
Zachary Hamaker ◽  
Eric Marberg ◽  
Brendan Pawlowski
Keyword(s):  
Power Series ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Pattern Avoidance ◽  
Dominance Order ◽  
Schubert Polynomials ◽  
Stanley Symmetric Functions ◽  
Skew Schur Functions

Abstract The involution Stanley symmetric functions$\hat{F}_y$ are the stable limits of the analogs of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words and are indexed by the involutions in the symmetric group. By construction, each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila–Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials.

scholarly journals Product of Stanley symmetric functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Nan Li
Keyword(s):  
Nous Avons ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Nous Obtenons ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Natural Expansion ◽  
Natural Way

International audience We study the problem of expanding the product of two Stanley symmetric functions $F_w·F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, and study the behavior of the expansion of $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. Nous étudions le problème de développement du produit de deux fonctions symétriques de Stanley $F_w·F_u$ en fonctions symétriques de Stanley de façon naturelle. Notre méthode consiste à considérer une fonction symétrique de Stanley comme un polynôme du Schubert stabilisè $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, et à étudier le comportement de développement de $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ en polynômes de Schubert lorsque $n$ augmente. Nous prouvons que cette développement se stabilise et donc nous obtenons une développement naturelle pour le produit de deux fonctions symétriques de Stanley. Dans le cas où l'une des permutations est Grassmannienne, nous avons une meilleure compréhension de cette stabilité.

MRI analysis and surgical treatment of Wassel Type IV duplicated thumbs

2021 ◽  
pp. 175319342098321
Author(s):  
Anyuan Wang ◽  
Jian Ding ◽  
Long Wang ◽  
Tinggang Chu ◽  
Zhipeng Wu ◽  
...  
Keyword(s):  
Type A ◽  
Surgical Reconstruction ◽  
Mri Findings ◽  
Level Of Evidence ◽  
Type D ◽  
Type Iv ◽  
Secondary Operations ◽  
Type C

We present the MRI findings for 39 Wassel Type IV duplicated thumbs in 38 patients. We found that MRI revealed the morphology of the cartilaginous connection between the thumb anlages and the location of the deviation corresponding to the classification of Horii, which allowed precise preoperative planning of corrective osteotomies. All 39 thumbs were available for follow-up after surgical reconstruction at a mean of 29 months (range 25 to 39). Four out of nine Horii Type A cases and all 12 Type B, as well as the six Type C and the six Type D cases, achieved good results according to the Tada scoring system. Five Type A cases achieved fair results with residual stiffness of the interphalangeal joint. No secondary operations were needed. We conclude that MRI proved useful in subclassifying Wassel Type IV duplicated thumbs and may aid in planning the osteotomies needed for their reconstruction. Level of evidence: IV

MPTP delta, a putative murine homolog of HPTP delta, is expressed in specialized regions of the brain and in the B-cell lineage

1993 ◽  
Vol 13 (9) ◽  
pp. 5513-5523
Author(s):  
K Mizuno ◽  
K Hasegawa ◽  
T Katagiri ◽  
M Ogimoto ◽  
T Ichikawa ◽  
...  
Keyword(s):  
B Cell ◽  
Cell Lines ◽  
Type A ◽  
Extracellular Domain ◽  
Fine Tuning ◽  
Leader Peptide ◽  
Reticular Nucleus ◽  
Cdna Clones ◽  
Protein Tyrosine ◽  
Type C

Protein tyrosine phosphatases (PTPs), together with protein tyrosine kinases (PTKs), are involved in the regulation of cell activation, growth, and differentiation. To further elucidate the fine tuning of cell growth and differentiation through tyrosine phosphorylation, we tried to isolate mouse receptor-type PTP (RPTP) cDNA clones by screening mouse brain cDNA libraries with mouse CD45 PTP domain probes under reduced-stringency conditions. Characterization of isolated cDNA clones for RPTP showed that the cytoplasmic region contains two tandem repeats of PTP domain of about 230 amino acids with intrinsic phosphatase activity. The extracellular region was composed of immunoglobulin (Ig)-like domains and fibronectin type III (FN-III)-like domains. The gene was highly homologous to human PTP delta (HPTP delta) and thus was named MPTP delta (murine counterpart of HPTP delta). The MPTP delta gene appeared to generate at least three species of mRNA, which differ in the composition of the extracellular domain: type A, one Ig-like and four FN-III-like domains; type B, one Ig-like and eight FN-III-like domains; and type C, three Ig-like and eight FN-III-like domains. Interestingly, the 5' untranslated region and the leader peptide of types A and B were completely different from those of type C. Northern (RNA) blot analysis demonstrated that brain, kidney, and heart cells express three mRNA species of about 7 kb. Antibody directed against part of the extracellular domain of type A MPTP delta recognized a 210-kDa protein in brain and kidney lysates. In situ hybridization of brain samples revealed that MPTP delta mRNA is present in the hippocampus, thalamic reticular nucleus, and piriform cortex, where some Src family PTKs have been also demonstrated to exist. Although MPTP delta mRNA was not detected in lymphoid tissues, all of the pre-B-cell lines tested and one of three B-cell lines tested expressed MPTP delta mRNA, whereas antibody-producing B-cell hybridomas and T-cell and macrophage lines did not. Finally, the MPTP delta locus was tightly linked to the brown (b) locus on mouse chromosome 4.

Abstract 19622: Trans-apical Off-pump Mitral Valve Repair With Neochords Implantation: One Year Clinical and Echocardiographic Follow-up

Circulation ◽  
2015 ◽  
Vol 132 (suppl_3) ◽  
Author(s):  
Andrea Colli ◽  
Laura Besola ◽  
Lorenzo Bagozzi ◽  
Erica Manzan ◽  
Eleonora Bizzotto ◽  
...  
Keyword(s):  
Mitral Valve ◽  
Type A ◽  
Mitral Valve Regurgitation ◽  
Disease Type ◽  
Reverse Remodeling ◽  
Off Pump ◽  
Single Centre Study ◽  
Type C ◽  
One Year

Introduction: TOP-MINI is a new micro invasive surgical procedure to treat degenerative mitral valve regurgitation due to flail/prolapse. Hypothesis: This prospective single centre study sought to assess the safety and effectiveness of the TOP-MINI procedure up to one year follow-up. Methods: Clinical and Echocardiographic outcomes were evaluated at 1, 3, 6 months and 1 year follow-up for all patients underwent TOP-MINI procedure from November 2013 to March 2015. Procedural success was defined as residual MR≤2+ at any time. Results: Sixty-one patients were treated during study period. One year survival was 96.7±2.3%. Freedom from MR>2+ is shown in figure 1 Panel A, Freedom from MR>2+ according to valve anatomy (Type A isolated P2 disease, Type B posterior multisegment disease, Type C anterior or bileaflet and/or calcified disease) is shown in Figure 1 Panel B. Freedom from MR>2+ according to STS risk profile is shown in Figure 2. The trend of Echocardiographic parameters is shown in Figure 3. Conclusions: TOP-MINI is a safe and effective procedure at 1 year FU. Residual MR is influenced by valve anatomy showing good results in Type A and B patients. Future techniques refinements are needed in order to improve outcomes of Type C patients. The lack of annuloplasty procedure does not influence negatively left ventricle reverse remodeling.

scholarly journals A surveillance of enteropathogens in piglets from birth to seven days of age in Brazil

2013 ◽  
Vol 33 (8) ◽  
pp. 963-969 ◽  
Author(s):  
Eduardo C. Cruz Junior ◽  
Felipe M. Salvarani ◽  
Rodrigo O.S. Silva ◽  
Marcos X. Silva ◽  
Francisco C.F. Lobato ◽  
...  
Keyword(s):  
Escherichia Coli ◽  
Clostridium Difficile ◽  
Type A ◽  
Final Diagnosis ◽  
Enterotoxigenic Escherichia Coli ◽  
Microbiological Test ◽  
E Coli ◽  
Isospora Suis ◽  
Real Importance ◽  
Type C

The purpose of the study was to evaluate the real importance of anaerobic enteropathogens and rotavirus in contrast to more common agents as cause of diarrhea in piglets within the first week of life. Sixty 1- to 7-day-old piglets, 30 diarrheic and 30 non-diarrheic (control), from 15 different herds were selected, euthanized and necropsied. Samples of the jejunum, ileum, colon, cecum and feces were collected from the piglets and analyzed to determine the presence of the following enteropathogens: enterotoxigenic Escherichia coli (ETEC), Clostridium perfringens types A and C, Clostridium difficile, rotavirus and Isospora suis. Among diarrheic piglets, 23.3% were positive for C. difficile, 70% for C. perfringens type A cpb2+, 14.3% for rotavirus and 10% for ETEC. Among non-diarrheic control piglets, 10% were positive for C. difficile, 76.7% for C. perfringens type A cpb2+, 0% for rotavirus, 3.3% for ETEC and 3.3% for I. suis. C. perfringens type C was not detected in any of the animals. Histological lesions characteristic of C. difficile, E. coli and rotavirus were observed. However, no C. perfringens type A suggestive lesions were detected. There was a positive correlation between mesocolon edema and the presence of C. difficile toxins. Although C. perfringens type A cpb2+ was the most frequently detected enteropathogen, there was no association between its presence and diarrhea or macro or microscopic changes. C. difficile and Rotavirus were the most relevant pathogens involved with neonatal diarrhea in this study, and histopathology associated with microbiological test proved to be the key to reach a final diagnosis.

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