User rekha biswal - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:21:13Z http://mathoverflow.net/feeds/user/21155 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125709/bijection-between-number-of-partitions-of-2n-satisfying-certain-conditions-with-n bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n Rekha Biswal 2013-03-27T10:39:28Z 2013-04-15T15:39:48Z <p>Suppose $\lambda = (\lambda_1,\lambda_2,.....,\lambda_k)$ is a partition of $2n$ where $n \in \mathbb N$ satisfying the following conditions:</p> <p>(1) $\lambda_{k} = 1$.</p> <p>(2) $\lambda_{i} - \lambda_{i+1} \leq 1$ for every $i\leq k-1$.</p> <p>(3) In the partition $\lambda$, the number of odd parts in odd places &amp; the number of odd parts in even places are equal. Here a part $\lambda_{i}$ is said to be in even place if ${i}$ is even, whereas $\lambda_{i}$ is said to be in odd place if ${i}$ is odd. $\lambda_{i}$ 's are called parts of $\lambda$ and $\lambda_{i}$ is called an odd part if it is odd &amp; is called even part if it is even.</p> <p>Now the question is to give a bijection between number of partitions of $2n$ satisfying the above conditions and number of partitions of $n$.</p> http://mathoverflow.net/questions/125708/multiplicity-of-a-weight-in-the-basic-representation-of-hatsl-2 multiplicity of a weight in the basic representation of $\hat{sl_2}$ Rekha Biswal 2013-03-27T10:17:00Z 2013-03-27T10:17:00Z <p>it is known by Weyl character formula that multiplicity of the weight $\wedge_{0}- n.\delta$ in the basic representation $L(\wedge_{0})$ of $\hat{sl_2}$is equal to the number of partitions of $n$.It is known that LS(Lakshmibai seshadri) paths of shape $\wedge_{0}$ gives a parametrization for the basis of $L(\wedge_{0})$.There is a paper of Yasmine B.Sanderson on dimensions of Demazure modules for rank two affine algebras where she has found L-S paths for the basic representation of $\hat{sl_2}$ explicitly. Here is the explicit description of paths. The L-S Paths of shape $\wedge_{0}$ are those paths $\pi = (\sigma,a)$ such that $\sigma:w_{n+k} > w_{n+k-1} > w_{n+k-2} >...> w_{n}$ where $w_i$ is the product of $i$ reflections $r_{0}.r_{1}.r_{0}.r_{1}...$ and $n,k \geq 0$ and $a:0 &lt; a_{n+k} &lt; a_{n+k-1} &lt; ...... &lt; a_{n+1} &lt; 1$ where $a_j.d_j\epsilon Z$ and $d_j = -j$ and $a_{i} \epsilon Q$ Now the question is how can one show that there are exactly $P(n)$ of L-S paths of shape $\wedge_{0}$ whose weight is $\wedge_{0} - n.\delta$ ?</p> http://mathoverflow.net/questions/123654/character-formula-for-demazure-modules character formula for demazure modules Rekha Biswal 2013-03-05T19:00:24Z 2013-03-05T20:39:42Z <p>Is there any character formula for demazure modules in arbitary kac moody settings which does not use demazure operators?</p> http://mathoverflow.net/questions/119948/conditions-under-which-a-l-s-path-is-undefined conditions under which a $(L-S)$ path is undefined. Rekha Biswal 2013-01-26T16:13:08Z 2013-01-26T16:13:08Z <p>Let $\pi$ be a Lakshmibai seshadri $(L-S)$ path of weight $\lambda$ .Let $B_{\pi}$ be the set of paths obtained from $\pi$ by applying the root operatos.It is known that the formal character of $B_{\pi}$ is equal to the character of simple module of highest weight $\lambda$ of a symmetrizable kac moody algebra $L$. $B_{\pi}$ is equal to the set of paths obtained from $\lambda$ by applying lowering operators corresponding to different simple roots of $L$. let $f^a_{\alpha_1}.f^b_{\alpha_2}...f^t_{\alpha_n}.\pi$ be a path in $B_{\pi}$.under what conditions on $a$,$b$,... $t$ the path $f^a_{\alpha_1}.f^b_{\alpha_2}...f^t_{\alpha_n}.\pi$ is undefined where $a$,$b$,$c$... $t$ are non negative integers?Is it possible to find such conditions at least for rank 2 case?</p> http://mathoverflow.net/questions/115364/poincare-series-of-subsets-of-affine-weyl-groups poincare series of subsets of affine weyl groups Rekha Biswal 2012-12-04T06:54:26Z 2012-12-04T06:54:26Z <p>It is known that if A is subset of reflection subgroup of an affine coxeter group W then the poincare series of minimal length coset representatives of A is a rational function.From this fact is it true that poincare series of A itself is a rational function with respect to the length function Of W?</p> http://mathoverflow.net/questions/115293/length-function-of-a-coxeter-group-with-respect-to-two-different-simple-systems-a length function of a coxeter group with respect to two different simple systems are equal or not? Rekha Biswal 2012-12-03T13:54:52Z 2012-12-03T19:51:20Z <p>Is there any relation between length function of a coxeter group with respect to two different simple systems as two simple systems are weyl conjugates of one another?</p> http://mathoverflow.net/questions/115295/poincare-series-of-subsets-of-hyperbolic-coxeter-groups-are-rational-or-not poincare series of subsets of hyperbolic coxeter groups are rational or not? Rekha Biswal 2012-12-03T14:00:13Z 2012-12-03T14:00:13Z <p>If W is a hyperbolic coxeter group &amp; A is a reflection subgroup of W &amp; W^A is the set of minimal length coset representatives of A in W then is the poincare series of W^A is rational?</p> http://mathoverflow.net/questions/112774/number-of-distinct-irreducible-representations-of-semisimple-lie-algebra-upto-iso number of distinct irreducible representations of semisimple lie algebra upto isomorphism Rekha Biswal 2012-11-18T17:30:08Z 2012-11-18T17:30:08Z <p>Prove that the number of distinct irreducible representations L(\lamda) of semisimple lie algebra of dimension less than or equal to "n",where n is an integer is finite.</p> http://mathoverflow.net/questions/111127/finding-highest-weight-of-dual-of-a-representation-of-a-semisimple-lie-algebra finding highest weight of dual of a representation of a semisimple lie algebra Rekha Biswal 2012-11-01T10:02:41Z 2012-11-02T12:19:04Z <p>If V is an irreducible representation of a semi simple lie algebra having highest weight λ then what will be the highest weight of the corresponding irreducible representation V∗ (Dual of V)?</p> http://mathoverflow.net/questions/125709/bijection-between-number-of-partitions-of-2n-satisfying-certain-conditions-with-n/125719#125719 Comment by Rekha Biswal Rekha Biswal 2013-03-29T12:56:31Z 2013-03-29T12:56:31Z @Wouter what is the meaning of bijection to OEIS A064174 here? http://mathoverflow.net/questions/125709/bijection-between-number-of-partitions-of-2n-satisfying-certain-conditions-with-n/125728#125728 Comment by Rekha Biswal Rekha Biswal 2013-03-28T16:07:01Z 2013-03-28T16:07:01Z @Fayers how is c_0 defined here? http://mathoverflow.net/questions/125709/bijection-between-number-of-partitions-of-2n-satisfying-certain-conditions-with-n Comment by Rekha Biswal Rekha Biswal 2013-03-27T17:52:25Z 2013-03-27T17:52:25Z @Barry oh Thanks .since you have already provided the link .I will take care of it from next time. http://mathoverflow.net/questions/125709/bijection-between-number-of-partitions-of-2n-satisfying-certain-conditions-with-n Comment by Rekha Biswal Rekha Biswal 2013-03-27T12:29:42Z 2013-03-27T12:29:42Z BY weyl character formula it is proved that multiplicity of the weight $\wedge_{0})- n.\delta$ in the basic representation of $\hat{sl_2}$ is eual to the number pf partitions of n.In paper of Kreiman,Lakshmibai,Magyar &amp; Weyman on standard basis of affine sl_n modules ,they have given a parametrization for the basis of $L(\wedge_{0})$ in terms of semi infinite wedge wedge product.from that parametrization it has been found that the multiplicity of weight $\wedge_{0}-n.\delta$ is exactly equal to the number of partitions $\lambda$ of 2n satisfying the above conditions mentioned in the question. http://mathoverflow.net/questions/125708/multiplicity-of-a-weight-in-the-basic-representation-of-hatsl-2 Comment by Rekha Biswal Rekha Biswal 2013-03-27T12:19:14Z 2013-03-27T12:19:14Z You can check the notations in Yasmine B.Sanderson 's paper on dimensions of demazure modules for rank two affine algebras in the journal Compositio Mathematica ,tome 101,n^0 2(1996),p.115-131 for the notations. http://mathoverflow.net/questions/115293/length-function-of-a-coxeter-group-with-respect-to-two-different-simple-systems-a/115333#115333 Comment by Rekha Biswal Rekha Biswal 2012-12-04T04:32:28Z 2012-12-04T04:32:28Z @Jim actually the question is as follows,how the length function of elements of weyl group are related with respect to two different simple systems? http://mathoverflow.net/questions/111127/finding-highest-weight-of-dual-of-a-representation-of-a-semisimple-lie-algebra/111228#111228 Comment by Rekha Biswal Rekha Biswal 2012-11-02T14:51:30Z 2012-11-02T14:51:30Z @Jim Humphreys thank you sir for your nice explanation. http://mathoverflow.net/questions/111127/finding-highest-weight-of-dual-of-a-representation-of-a-semisimple-lie-algebra/111131#111131 Comment by Rekha Biswal Rekha Biswal 2012-11-01T12:42:05Z 2012-11-01T12:42:05Z @Sasha could you please explain how -w(lamda) will be the highest weight? http://mathoverflow.net/questions/90235/irreducible-elements Comment by Rekha Biswal Rekha Biswal 2012-03-05T00:05:50Z 2012-03-05T00:05:50Z I want a simple elementary proof of this, not using the ideals.