Characters for Projective Modules in the BGG Category O for General Linear Lie Superalgebras

We determine the Verma multiplicities and the characters of projective modules for atypical blocks in the BGG Category O for the general linear Lie superalgebras $\frak{gl}(2|2)$ and $\frak{gl}(3|1)$. We then explicitly determine the composition factor multiplcities of Verma modules in the atypicality 2 block of $\frak{gl}(2|2)$

Explicit Demazure character formula for negative dominant characters

In this paper, we prove that for any semisimple simply connected algebraic group $G$, for any regular dominant character $\lambda$ of a maximal torus $T$ of $G$ and for any element $\tau$ in the Weyl group $W$, the character $e^{\rho}\cdot char(H^{0}(X(\tau), \mathcal{L}_{\lambda-\rho}))$ is equal to the sum $\sum_{w\leq \tau}char(H^{l(w)}(X(w),\mathcal{L}_{-\lambda}))^{*})$ of the characters of dual of the top cohomology modules on the Schubert varieties $X(w)$, $w$ running over all elements satisfying $w\leq \tau$. Using this result, we give a basis of the intersection of the Kernels of the Demazure operators $D_{\alpha}$ using the sums of the characters of $H^{l(w)}(X(w),\mathcal{L}_{-\lambda})$, where the sum is taken over all elements $w$ in the Weyl group $W$ of $G$.Comment: 11 page

Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.Comment: 19 page

Association Rule Pruning based on Interestingness Measures with Clustering

Association rule mining plays vital part in knowledge mining. The difficult task is discovering knowledge or useful rules from the large number of rules generated for reduced support. For pruning or grouping rules, several techniques are used such as rule structure cover methods, informative cover methods, rule clustering, etc. Another way of selecting association rules is based on interestingness measures such as support, confidence, correlation, and so on. In this paper, we study how rule clusters of the pattern Xi - Y are distributed over different interestingness measures.Comment: International Journal of Computer Science Issues, IJCSI Volume 6, Issue 1, pp35-43, November 200

Nondegeneracy for Quotient Varieties under Finite Group Actions

We prove that for an abelian group $G$ of order $n$ the morphism $\varphi\colon \mathbf{P}(V^*)\longrightarrow \mathbf{P} ((\mathrm{sym}^n V^*)^G)$ defined by $\varphi([f]) = [\prod_{\sigma\in G} \sigma \cdot f ]$ is nondegenerate for every finite-dimensional representation $V$ of $G$ if and only if either $n$ is a prime number or $n=4$.Comment: 8 page

Projective normality of finite group quotients and EGZ theorem

In this note, we prove that for any finite dimensional vector space $V$ over $\mathbb {C}$, and for a finite cyclic group $G$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$ by a method using toric variety, and deduce the EGZ theorem as a consequence.Comment: 3 page

Torus quotients of homogeneous spaces-minimal dimensional Schubert Variety admitting semi-stable points

In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. In this paper, we also describe, for any semi-simple simply connected algebraic group $G$ and for any Borel subgroup $B$ of $G$, all Coxeter elements $\tau$ for which the Schubert variety $X(\tau)$ admits a semistable point for the action of the torus $T$ with respect to a non-trivial line bundle on $G/B$

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation $V$of the Weyl group $W$ of a semi-simple algebraic group of type $A_n, B_n, C_n, D_n, F_4$ and $G_2$ over $\mathbb C$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V$. We also prove that for any finite group $G$ and for any finite dimentional representation $V$ over $\mathbb C$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes n!}$ as a consequence.Comment: 10 page

Role of Interestingness Measures in CAR Rule Ordering for Associative Classifier: An Empirical Approach

Associative Classifier is a novel technique which is the integration of Association Rule Mining and Classification. The difficult task in building Associative Classifier model is the selection of relevant rules from a large number of class association rules (CARs). A very popular method of ordering rules for selection is based on confidence, support and antecedent size (CSA). Other methods are based on hybrid orderings in which CSA method is combined with other measures. In the present work, we study the effect of using different interestingness measures of Association rules in CAR rule ordering and selection for associative classifier

Syzygies of some GIT quotients

Let $X$ be flat scheme over $\mathbb{Z}$ such that its base change, $X_p$, to $\bar{\mathbb{F}}_p$ is Frobenius split for all primes $p$. Let $G$ be a reductive group scheme over $\mathbb{Z}$ acting on $X$. In this paper, we prove a result on the $N_p$ property for line bundles on GIT quotients of $X_{\mathbb{C}}$ for the action of $G_{\mathbb{C}}$. We apply our result to the special cases of (1) an action of a finite group on the projective space and (2) the action of a maximal torus on the flag variety of type $A_n$.Comment: 11 pages; improved bounds in main results; new references adde