Conjugacy classes and growth conditions

AbstractIn this paper we show using a purely combinatorial argument that a finitely generated infinite group such that fE(n)⩽ans, where a is a constant, admits for every ϵ a sequence {gi,ϵ} of non-unit elements whose centralizer contains more than i1/2−ϵ elements of length less than i. Of course, the interest of this result is in the fact that it excludes the possibility that the group is a pure torsion group, since otherwise the existence of the sequence {gi,ϵ} is obvious. As an application of this result, we show that, in the case where r<3/2, there exists an element whose centralizer has finite index in G

On the combinatorial invariance of Kazhdan–Lusztig polynomials

AbstractIn this paper, we solve the conjecture about the combinatorial invariance of Kazhdan–Lusztig polynomials for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. We also obtain explicit formulas for the R-polynomials and for the Kazhdan–Lusztig polynomials associated with any interval of length 5 in any Coxeter group, showing in particular what they look like in the symmetric group

Lattice paths, lexicographic correspondence and Kazhdan–Lusztig polynomials

AbstractIn this paper we give a new closed formula for the Kazhdan–Lusztig polynomials of finite Coxeter groups and affine Weyl groups. This formula is computationally more efficient than any existing one, and it conjecturally holds for all Coxeter groups

Comparing and characterizing some constructions of canonical bases from Coxeter systems

The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a "standard basis" indexed by the elements of $W$ and a "bar involution" given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan-Lusztig basis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan have defined a representation of a modified Iwahori-Hecke algebra on the free $\mathbb{Z}[v,v^{-1}]$-module generated by the set of twisted involutions in $W$, and shown that this module has a unique pre-canonical structure satisfying a certain compatibility condition, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan-Lusztig basis. One can modify the parameters defining Lusztig and Vogan's module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this fashion, and explain the relationships between their resulting canonical bases. While some of these canonical bases are related in a trivial fashion to Lusztig and Vogan's construction, others appear to have no simple relation to what has been previously studied. Along the way, we also clarify the differences between Webster's notion of a canonical basis and the related concepts of an IC basis and a $P$-kernel.Comment: 32 pages; v2: additional discussion of relationship between canonical bases, IC bases, and P-kernels; v3: minor revisions; v4: a few corrections and updated references, final versio

Kazhdan-Lusztig polynomials, tight quotients and Dyck superpartitions

We give an explicit combinatorial product formula for the parabolic Kazhdan–Lusztig polynomials of the tight quotients of the symmetric group. This formula shows that these polynomials are always either zero or a monic power of q and implies the main result in [F. Brenti, Kazhdan–Lusztig and R-polynomials, Youngʼs lattice, and Dyck partitions, Pacific J. Math. 207 (2002) 257–286] on the parabolic Kazhdan–Lusztig polynomials of the maximal quotients. The formula depends on a new class of superpartitions

Combinatorics of BB-orbits and Bruhat--Chevalley order on involutions

Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $\sigma$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $\Omega_{\sigma}\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of rook placements and is dual to A. Melnikov's results on $B$-orbits on $\mathfrak{u}$. Using results of F. Incitti, we also prove that this partial order coincides with the restriction of the Bruhat--Chevalley order to the set of involutions.Comment: 27 page

Prediction of the outcome of preoperative chemotherapy in breast cancer using DNA probes that provide information on both complete and incomplete responses

<p>Abstract</p> <p>Background</p> <p>DNA microarray technology has emerged as a major tool for exploring cancer biology and solving clinical issues. Predicting a patient's response to chemotherapy is one such issue; successful prediction would make it possible to give patients the most appropriate chemotherapy regimen. Patient response can be classified as either a pathologic complete response (PCR) or residual disease (NoPCR), and these strongly correlate with patient outcome. Microarrays can be used as multigenic predictors of patient response, but probe selection remains problematic. In this study, each probe set was considered as an elementary predictor of the response and was ranked on its ability to predict a high number of PCR and NoPCR cases in a ratio similar to that seen in the learning set. We defined a valuation function that assigned high values to probe sets according to how different the expression of the genes was and to how closely the relative proportions of PCR and NoPCR predictions to the proportions observed in the learning set was. Multigenic predictors were designed by selecting probe sets highly ranked in their predictions and tested using several validation sets.</p> <p>Results</p> <p>Our method defined three types of probe sets: 71% were mono-informative probe sets (59% predicted only NoPCR, and 12% predicted only PCR), 25% were bi-informative, and 4% were non-informative. Using a valuation function to rank the probe sets allowed us to select those that correctly predicted the response of a high number of patient cases in the training set and that predicted a PCR/NoPCR ratio for validation sets that was similar to that of the whole learning set. Based on DLDA and the nearest centroid method, bi-informative probes proved more successful predictors than probes selected using a t test.</p> <p>Conclusion</p> <p>Prediction of the response to breast cancer preoperative chemotherapy was significantly improved by selecting DNA probe sets that were successful in predicting outcomes for the entire learning set, both in terms of accurately predicting a high number of cases and in correctly predicting the ratio of PCR to NoPCR cases.</p

On the Commutative Equivalence of Context-Free Languages

The problem of the commutative equivalence of context-free and regular languages is studied. In particular conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated

Promoter hypermethylation of HS3ST2, SEPTIN9 and SLIT2 combined with FGFR3 mutations as a sensitive/specific urinary assay for diagnosis and surveillance in patients with low or high-risk non-muscle-invasive bladder cancer

International audienceBackgroundNon-muscle-invasive bladder cancer (NMIBC) is a high incidence form of bladder cancer (BCa), where genetic and epigenetic alterations occur frequently. We assessed the performance of associating a FGFR3 mutation assay and a DNA methylation analysis to improve bladder cancer detection and to predict disease recurrence of NMIBC patients.MethodsWe used allele specific PCR to determine the FGFR3 mutation status for R248C, S249C, G372C, and Y375C. We preselected 18 candidate genes reported in the literature as being hypermethylated in cancer and measured their methylation levels by quantitative multiplex-methylation specific PCR. We selected HS3ST2, SLIT2 and SEPTIN9 as the most discriminative between control and NMIBC patients and we assayed these markers on urine DNA from a diagnostic study consisting of 167 NMIBC and 105 controls and a follow-up study consisting of 158 NMIBC at diagnosis time’s and 425 at follow-up time. ROC analysis was performed to evaluate the diagnostic accuracy of each assay alone and in combination.ResultsFor Diagnosis: Using a logistic regression analysis with a model consisting of the 3 markers’ methylation values, FGFR3 status, age and known smoker status at the diagnosis time we obtained sensitivity/specificity of 97.6 %/84.8 % and an optimism-corrected AUC of 0.96. With an estimated BCa prevalence of 12.1 % in a hematuria cohort, this corresponds to a negative predictive value (NPV) of 99.6 %. For Follow-up: Using a logistic regression with FGFR3 mutation and the CMI at two time points (beginning of the follow-up and current time point), we got sensitivity/specificity/NPV of 90.3 %/65.1 %/97.0 % and a corrected AUC of 0.84. We also tested a thresholding algorithm with FGFR3 mutation and the two time points as described above, obtaining sensitivity/specificity/NPV values of, respectively, 94.5 %/75.9 %/98.5 % and an AUC of 0.82.ConclusionsWe showed that combined analysis of FGFR3 mutation and DNA methylation markers on urine can be a useful strategy in diagnosis, surveillance and for risk stratification of patients with NMIBC. These results provide the basis for a highly accurate noninvasive test for population screening and allowing to decrease the frequency of cystoscopy, an important feature for both patient quality of life improvement and care cost reduction