Nov
19 |
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Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?
@AbdelmalekAbdesselam : the general setup is that if you have a system of parameters $P$ (or some other well-defined set of generators of a (sub)ring - I'd have to check Derksen-Kemper to see exactly which one) of the corresponding invariant ring, and evaluate your graph (as a vector of 0s and 1s) on this system, you will get a unique fingerprint, distinguishing nonisomorphic graphs. Pouzet's conjecture was that certain set $P'$ is generating the same ring as $P$, but this turned out to be false. |

Nov
19 |
comment |
Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?
@AbdelmalekAbdesselam : what do you mean by "doesn't sound good"? That they make an error? IIRC, these conjectures were about particular generating sets for rings, not about general setup. The latter is sound, I think. |

Nov
13 |
comment |
Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?
it's known how to search for generators for invariant rings of permutation groups, of which we have an example here (e.g. take orbit sums of monomials, up to certain degree). The problem is that such a set will be huge, and will have huge degrees. |

Nov
12 |
comment |
@AbdelmalekAbdesselam : having a "good" set of generators for such a ring (or only a system of parameters for such a ring) would allow one to solve the isomorphism problem for bipartite graphs (which is as hard as for general graphs -- cf. e.g. en.wikipedia.org/wiki/…). How a similar idea works for general graphs, is explained in the book I cited in the answer. |

Nov
3 |
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What does Kqn means here?
just a typo, perhaps? |

Oct
27 |
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Weights on cyclic orderings
we (and Woodall, and Kleitman) define the graph on these $(n-1)!$ elements of $S_n$, so that two such elements are adjacent if one needs to swap two adjacent positions to get one from the other. |

Oct
27 |
comment |
How to find solutions for four polynomial equations with four unknown variables using Resultant Theory
you should also specify whether you look for complex solutions, or only for real solutions, or actually it happens over some other field... |

Oct
26 |
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Strongly connected graph and the eigenvalues of the laplacian matrix
so you see, the laplacian has one eigenvalue 0, and 2 conjugate complex eignevalues, each with multiplicity 3. |

Oct
26 |
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Strongly connected graph and the eigenvalues of the laplacian matrix
sure, see the edited answer |

Oct
14 |
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notation for vector product in the space
@WillieWong surely, $\times$ is overused, but using it on vectors does not clash with using it on scalars, or on sets (or other things one can Cartesian-multiply). As opposed to two different meanings of $\wedge$ for vectors... |

Oct
13 |
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notation for vector product in the space
of course in the exterior algebra of $\mathbb{R}^3$ one has that $a\wedge b\wedge c$ is a scalar, so this identification cannot go too far. So you tell undergrads that they shouldn't think of $a\wedge b\wedge c$, and then in multivariate calculus in $\mathbb{R}^n$ you all of a sudden start talking about $a\wedge b\wedge c$? |

Oct
13 |
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notation for vector product in the space
Exterior product is associative, and the vector product is not. Thus I don't get how they can be identified... |

Oct
7 |
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On the theory of infinite extraspecial $p$-groups
Philip, if $a^2=b^2=(ab)^2=1$ then $ab=ba$, and you get an abelian group. |

Oct
3 |
comment |
asymptotic for the number of involutions in GL(n,2)
OK, I think I'm being lazy, I should just sit down and calculate :-) |

Oct
3 |
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asymptotic for the number of involutions in GL(n,2)
Well, I only did such things for ordinary hypergeometric functions, not for basic hypergeometric ones, like here. Any place to look for these things? |

Oct
3 |
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asymptotic for the number of involutions in GL(n,2)
I added a formula for each of [n/2] terms (plus the term for k=0, which should be 1, assuming $|GL_0(k)|=1$) in the question, but it does not look routine... |

Oct
1 |
comment |
Name for an operation on matrices?
what are these $B_{j[1]}$, $B_{j[2]}$, etc? |

Oct
1 |
comment |
Computer algebra system for Weyl algebra computations
you can run singular on cloud.sagemath.com |

Sep
30 |
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asymptotic for the number of involutions in GL(n,2)
John, this paper talks a lot about solutions of $X^2=0$ in upper triangular matrices over a finite field, but does not mention $GL$. |

Sep
30 |
comment |
asymptotic for the number of involutions in GL(n,2)
yes I did. No such sequence is known, apparently. |