User melanie matchett wood - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:44:55Z http://mathoverflow.net/feeds/user/2267 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96705/computer-package-for-representation-theory-of-the-symmetric-group Computer package for representation theory of the symmetric group Melanie Matchett Wood 2012-05-11T20:29:27Z 2012-05-17T01:32:38Z <p>Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):</p> <p>(1) $V \otimes W$</p> <p>(2) $S_\lambda V$, where $S_\lambda$ is a Schur functor, or even just $\wedge^s V$,</p> <p>where $V$ and $W$ are input as sums of irreducible representations, i.e. by partitions with coefficients, and output in the same format? </p> http://mathoverflow.net/questions/45664/symmetric-power-and-gauss-quadratic-form/72933#72933 Answer by Melanie Matchett Wood for Symmetric Power and Gauss quadratic form Melanie Matchett Wood 2011-08-15T16:12:10Z 2011-08-15T16:12:10Z <p>Forms $ax^2+bxy+cy^2$ for which $a,b,c,\in \mathbb Z$ and $b$ is even correspond to the symmetric submodule of $(\mathbb Z^2)^{\otimes 2}$, i.e. the $S_2$ fixed elements of $(\mathbb Z x\oplus \mathbb Z y)^{\otimes 2}$.</p> <p>Forms $ax^2+bxy+cy^2$ for which $a,b,c,\in \mathbb Z$ correspond to the symmetric quotient of $(\mathbb Z^2)^{\otimes 2}$, i.e. the quotient of $(\mathbb Z x\oplus \mathbb Z y)^{\otimes 2}$ by elements $a-\sigma(a)$ where $\sigma$ is the transposition of what is left and right of the tensor.</p> http://mathoverflow.net/questions/18609/binary-quadratic-forms-in-characteristic-2/72932#72932 Answer by Melanie Matchett Wood for Binary Quadratic Forms in Characteristic 2 Melanie Matchett Wood 2011-08-15T16:04:10Z 2011-08-15T16:04:10Z <p>Classes of binary quadratic forms over any commutative ring R (with no conditions on characteristic, PID, etc.) correspond to certain modules over quadratic extensions of R. This theory is completely general also in the forms it covers, i.e. covers non-primitive and discriminant zero forms. We can then restrict our attention to the cases of interest. </p> <p>For example, when R is an integral domain and the binary quadratic form has non-zero discriminant, the modules associated to the forms are exactly the ideal classes of the quadratic extension. </p> <p>The construction is as follows, simplied for your case of interest. From a form $ax^2+bxy+cy^2$, we form a quadratic $R$-algebra $Q:=R[\tau]/(\tau^2+b\tau+ac)$ and a module $M =Rx\oplus Ry$ with $\tau x=-cy-bx$ and $\tau y=ax$. Given a quadratic algebra and ideal, we can pick an $R$ basis $x,y$ of the ideal and then shift a generator $\tau$ of the quadratic algebra as necessary so that $\tau y$ is a multiple of $x$. Then you can simply read off the inverse map.</p> <p>What I've given above is the case when the binary quadratic form, quadratic algebra, and module are free over $R$. For general $R$, they will only be locally free, and the above gives a construction locally on free patches. There are also global, coordinate-free descriptions of the correspondence between binary quadratic forms and modules over quadratic algebras. For the details I refer you to my paper "Gauss composition over an arbitrary base" <a href="http://www.sciencedirect.com/science/article/pii/S0001870810003257" rel="nofollow">http://www.sciencedirect.com/science/article/pii/S0001870810003257</a></p> http://mathoverflow.net/questions/7625/hypercohomology-of-a-dg-algebra Hypercohomology of a dg-algebra Melanie Matchett Wood 2009-12-03T00:33:02Z 2009-12-03T19:23:25Z <p>Can someone give me a reference (note I am looking for a reference and not a proof) for the following:</p> <p>If a complex $C$ has a dg-algebra structure, then the hypercohomology $H^0R\pi_*C$ has an algebra structure, and if $M$ is a dg-module for $C$, then $H^0R\pi_*M$ is a module under $H^0R\pi_*C$. (Here I am thinking of $C$ as a complex of sheaves on some scheme $S$ with $\pi : S \rightarrow T$, but this should just be a fact of homological algebra.)</p> http://mathoverflow.net/questions/96705/computer-package-for-representation-theory-of-the-symmetric-group/96709#96709 Comment by Melanie Matchett Wood Melanie Matchett Wood 2012-05-11T21:30:36Z 2012-05-11T21:30:36Z I just looked at this, and it seems &quot;itensor&quot; will do (1), but I am not sure if there is a function there for (2) (but there could be and I am missing it because I don't know how to translate schur functors, or even exterior products, into symmetric polynomial language). http://mathoverflow.net/questions/72679/if-the-discriminant-of-a-binary-quadratic-form-has-high-valuation-is-the-form-a Comment by Melanie Matchett Wood Melanie Matchett Wood 2011-08-24T16:16:32Z 2011-08-24T16:16:32Z Example: $a=t$ and $b=2us+2t$ and $c=2us+t$. The discriminant is $4u^2s^2$. This form is not congruent to a square modulo $u^2$ http://mathoverflow.net/questions/72679/if-the-discriminant-of-a-binary-quadratic-form-has-high-valuation-is-the-form-a Comment by Melanie Matchett Wood Melanie Matchett Wood 2011-08-11T19:02:04Z 2011-08-11T19:02:04Z If we take Jordan's example $a = s^2 + pust$, $b = 2st + put^2$, $c = t^2$, then the discriminant is $p^2u^2t^4$, which has $u$-adic valuation $2$ (which is the relevant one---the $2$ or $p$ is a red herring).