arXiv subjects
Publications and source records attributed to Velmurugan S.
We determine the minimal polynomial of each element of the double cover $G$ of the symmetric or alternating group in every irreducible spin representation of $G$.
We determine precisely when the branching coefficients arising from the restriction of irreducible representations of the symmetric group $S_n$ to the dihedral subgroup $D_n$ are nonzero, and we establish uniform linear lower bounds outside a finite exceptional family. As a consequence, we recover and substantially generalize known positivity results for cyclic subgroups $C_n \leq S_n$. Analogous results are obtained for the alternating group $A_n$.
A Kronecker coefficient is the multiplicity of an irreducible representation of a finite group $G$ in a tensor product of irreducible representations. We define Kronecker Hecke algebras and use them as a tool to study Kronecker coefficients in finite groups. We show that the number of simultaneous conjugacy classes in a finite group $G$ is equal to the sum of squares of Kronecker coefficients, and the number of simultaneous conjugacy classes that are closed under elementwise inversion is the sum of Kronecker coefficients weighted by Frobenius-Schur indicators. We use these tools to investigate which finite groups have multiplicity-free tensor products. We introduce the class of doubly real groups, and show that they are precisely the real groups which have multiplicity-free tensor products. We show that non-Abelian groups of odd order, non-Abelian finite simple groups, and most finite general linear groups do not have multiplicity-free tensor products.
In this short note, we classify pairs of conjugacy classes of the symmetric group such that any non-linear irreducible character of the symmetric group vanishes on at least one of them.
Let $ (\rho, V) $ be an irreducible representation of the symmetric group $ S_n$ (or the alternating group $ A_n$), and let $ g $ be a permutation on $n$ letters with each of its cycle lengths divides the length of its largest cycle. We describe completely the minimal polynomial of $\rho(g)$, showing that, in most cases, it equals $x^{o(g)} - 1 $, with a few explicit exceptions. As a by-product, we obtain a new proof (using only combinatorics and representation theory) of a theorem of Swanson that gives a necessary and sufficient condition for the existence of a standard Young tableau of a given shape and major index $r \ \text{mod} \ n$, for all $r$. Thereby, we give a new proof of a celebrated result of Klyachko on Lie elements in a tensor algebra, and of a conjecture of Sundaram on the existence of an invariant vector for $n$-cycles. We also show that for elements $g$ in $S_n$ or $A_n$ of even order, in most cases, $\rho(g)$ has eigenvalue $-1$, with a few explicit exceptions.
We determine the eigenvalues with multiplicity of each element of an alternating group in any irreducible representation. This is equivalent to determining the decomposition of cyclic representations of alternating groups into irreducibles. We characterize pairs $(w, V)$, where $w$ is an element and $V$ is an irreducible representation of an alternating group such that $w$ admits a non-zero invariant vector in $V$. We also establish large new families of global conjugacy classes for alternating groups, thereby giving a new proof of a result of Heide and Zalessky on the existence of such classes.
The covering number of a non-linear character $χ$ of a finite group $G$ is the least positive integer $k$ such that every irreducible character of $G$ occurs in $χ^k$. We determine the covering numbers of irreducible characters of the symmetric group $S_n$ indexed by certain two-row partitions (and their conjugates), namely $(n-2,2)$ and $((n+1)/2, (n-1)/2)$ when $n$ is odd. We also determine the covering numbers of irreducible characters indexed by certain hook-partitions (and their conjugates), namely $(n-2,1^2)$, the almost self-conjugate hooks $(n/2+1, 1^{n/2-1})$ when $n$ is even, and the self-conjugate hooks $((n+1)/2, 1^{(n-1)/2})$ when $n$ is odd.
The algebra of symmetric functions contains several interesting families of symmetric functions indexed by integer partitions or skew partitions. Given a sequence $\{u_n\}$ of symmetric functions taken from one of these families such that $u_n$ is homogeneous of degree $n$, we provide necessary and sufficient conditions for the sequence to form a system of algebraically independent generators for the algebra of symmetric functions.
We determine when a permutation with cycle type $μ$ admits a non-zero invariant vector in the irreducible representation $V_λ$ of the symmetric group. We find that a majority of pairs $(λ,μ)$ have this property, with only a few simple exceptions.