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Author Archives: limsup
Schreier-Sims Algorithm
Introduction Throughout this article, we let G be a subgroup of generated by a subset We wish to consider the following questions. Given A, how do we compute the order of G? How do we determine if an element lies in G? Assuming , how … Continue reading
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Tagged group actions, group theory, permutations, programming, rubik's cube, schreier-sims, symmetries
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Primality Tests III
Solovay-Strassen Test This is an enhancement of the Euler test. Be forewarned that it is in fact weaker than the Rabin-Miller test so it may not be of much practical interest. Nevertheless, it’s included here for completeness. Recall that to … Continue reading
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Tagged cryptography, elementary, jacobi symbol, legendre symbol, number theory, primality tests, primes, programming
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Primality Tests II
In this article, we discuss some ways of improving the basic Fermat test. Recall that for Fermat test, to test if n is prime, one picks a base a < n and checks if We also saw that this method would utterly fail … Continue reading
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Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
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Primality Tests I
Description of Problem The main problem we wish to discuss is as follows. Question. Given n, how do we determine if it is prime? Prime numbers have opened up huge avenues in theoretical research – the renowned Riemann Hypothesis, for … Continue reading
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Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
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Polynomials and Representations XXXIX
Some Invariant Theory We continue the previous discussion. Recall that for we have a -equivariant map which induces an isomorphism between the unique copies of in both spaces. The kernel Q of this map is spanned by for various fillings T with shape and entries … Continue reading
Polynomials and Representations XXXVIII
Determinant Modules We will describe another construction for the Schur module. Introduce variables for . For each sequence we define the following polynomials in : Now given a filling T of shape λ, we define: where is the sequence of entries from the … Continue reading
Polynomials and Representations XXXVII
Notations and Recollections For a partition , one takes its Young diagram comprising of boxes. A filling is given by a function for some positive integer m. When m=d, we will require the filling to be bijective, i.e. T contains {1,…,d} and each element occurs exactly … Continue reading
Polynomials and Representations XXXVI
V(λ) as Schur Functor Again, we will denote throughout this article. In the previous article, we saw that the Schur-Weyl duality can be described as a functor: given a -module M, the corresponding -module is set as Definition. The construction is … Continue reading
Polynomials and Representations XXXV
Schur-Weyl Duality Throughout the article, we denote for convenience. So far we have seen: the Frobenius map gives a correspondence between symmetric polynomials in of degree d and representations of ; there is a correspondence between symmetric polynomials in and polynomial … Continue reading
Polynomials and Representations XXXIV
Twisting From the previous article, any irreducible polynomial representation of is of the form for some such that is the Schur polynomial . Now given any analytic representation V of G, we can twist it by taking for an integer k. Then: Twisting the irrep … Continue reading
Mathematics and Such 