arXiv subjects

Gurmail Singh

Publications and source records attributed to Gurmail Singh.

3 recordsLinked to original sources

arXiv2017

Classification of Homogeneous Fourier Matrices

Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group $SL_2(\mathbb{Z})$. In this paper, we show that there is a one-to-one correspondence between Fourier matrices associated to modular data and self-dual $C$-algebras that satisfy a certain condition. Also, we prove that a homogenous $C$-algebra arising from a Fourier matrix has all the degrees equal to $1$.

math.RA↗

arXiv2016

Classification of non-homogeneous Fourier matrices associated with modular data up to rank 5

Modular data is an important topic of study in rational conformal field theory. A modular datum defines finite dimensional representations of the modular group $\mbox{SL}_2(\mathbf{Z})$. For every Fourier matrix in a modular datum there exists an Allen matrix obtained from the Fourier matrix after dividing each its row with the first entry of that row. In this paper, we classify the non-homogenous Fourier matrices and non-homogenous Allen matrices up to rank $5$. The methods developed here are useful for the classification of the matrices of higher ranks. Also, we establish some results that are helpful in recognizing the $C$-algebras not arising from Allen matrices by just looking at the character table of the $C$-algebra, in particular, the first row of the character table.

math.RA↗

arXiv2016

The status of the Zassenhaus conjecture for small groups

We identify all small groups of order up to 288 in the GAP Library for which the Zassenhaus conjecture on rational conjugacy of units of finite order in the integral group ring cannot be established by an existing method. The groups must first survive all theoretical sieves and all known restrictions on partial augmentations (the HeLP$^+$ method). Then two new computational methods for verifying the Zassenhaus conjecture are applied to the unresolved cases, which we call the quotient method and the partially central unit construction method. To the cases that remain we attempt an assortment of special arguments available for units of certain orders and the lattice method. In the end, the Zassenhaus conjecture is verified for all groups of order less than 144 and we give a list of all remaining cases among groups of orders 144 to 287.

math.RA↗