Searcharxiv⌕ Search

arXiv subjects

Sagar Shrivastava

Publications and source records attributed to Sagar Shrivastava.

5 recordsLinked to original sources
arXiv2023

Relative Weyl Character formula, Relative Pieri formulas and Branching rules for Classical groups

We give alternate proofs of the classical branching rules for highest weight representations of a complex reductive group $G$ restricted to a closed regular reductive subgroup $H$, where $(G,H)$ consist of the pairs $(GL(n+1),GL(n))$, $ (Spin(2n+1), Spin(2n)) $ and $(Sp(2n),Sp(2)\times Sp(2n-2))$. Our proof is essentially a long division. The starting point is a relative Weyl character formula and our method is an inductive application of a relative Pieri formula. We also give a proof of the branching rule for the case of $ (Spin(2n), Spin(2n-1))$, by a reduction to the case of $(GL(n),GL(n-1))$.

math.RT↗
arXiv2023

Littlewood-Richardson coefficients as a signed sum of Kostka numbers

Littlewood-Richardson (LR) coefficients and Kostka Numbers appear in representation theory and combinatorics related to $GL_n$. It is known that Kostka numbers can be represented as special Littlewood-Rischardson coefficient. In this paper, we show how one can represent LR coefficient as a signed sum of Kostka numbers, and use the formulation to give a polynomial time algorithm for the same, hence showing that they belong to the same class of decision problems. As a corollary, we will prove Steinberg's formula using Kostant's partition function.

math.CO↗
arXiv2016

A Dedekind Domain with Nontrivial Class Group

Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of real-valued continuous functions on a closed interval like $[0,1]$ behaves similarly to the corresponding ring of complex-valued functions; they depend only on the topology of $[0,1]$. The ring $\mathbf{R}[X,Y]/(X^2+Y^2-1)$ of real-valued polynomial functions on the unit circle is not a unique factorization domain - witness the equation $$\cos^2(t) = (1+ \sin(t))(1- \sin(t)).$$ On the other hand, the ring $\mathbf{C}[X,Y]/(X^2+Y^2-1) \cong \mathbf{C}[X+iY, 1/(X+iY)]$ is a principal ideal domain. Again, the rings of convergent power series (over either of these fields) with radius of convergence larger than some number $\rho$ is a Euclidean domain (and hence, a principal ideal domain) - this can be seen by using for a Euclidean "norm" function, the function which counts zeroes (with multiplicity) in the disc $|z| \leq \rho$. In this note, we consider the rings $C_{an}(S^1;\mathbf{R})$ of real-analytic functions on the unit circle $\mathit{S}^1$ which are real-valued and the corresponding ring $C_{an}(S^1; \mathbf{C})$ of analytic functions that are complex-valued. We will see that the latter is a principal ideal domain while the former is a Dedekind domain which is not a principal ideal domain - the class group having order $2$.

math.RA↗
arXiv2016

Ring Of Real Analytic Functions on $[0,1]$

We consider the ring of real analytic functions defined on $[0,1]$, i.e. $$C^ω[0,1] =\lbrace f :[0,1] \longrightarrow \mathbb{R} | f \text{ is analytic on } [0,1]\rbrace$$ In this article, we explore the nature of ideals in this ring. It is well known that the ring $C[0,1]$ of real valued continuous functions on $[0,1]$ has precisely the following maximal ideals: $$\text{For } γ\in [0,1], M_γ := \lbrace f \in C[0,1] | f(γ) =0\rbrace$$ It has been proved that each such $M_γ$ is infinitely generated, in-fact uncountably generated. Observe that $C^ω[0,1]$ is a subring of $C[0,1]$ We prove that for any $γ$ in $[0,1]$, the contraction $M^ω_γ$ of $M_γ$ under the natural inclusion of $C^ω[0,1]$ in $C[0,1]$ is again a maximal ideal (of $C^ω[0,1]$ ), and these are precisely all the maximal ideals of $C^ω[0,1]$. Next we prove that each $M^ω_γ$ is principal (though $M_γ$ is uncountably generated). Surprisingly, this forces all the ideals of the ring $C^ω[0,1]$ to be singly generated, i.e. $C^ω[0,1]$ is a PID.

math.AC↗