[Paper Review] The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces

Richard Delanghe, Roman Lávička+1 more|arXiv (Cornell University)|Dec 22, 2010

Mathematical Analysis and Transform Methods22 references4 citations

TL;DR

This paper constructs explicit orthogonal Gelfand-Tsetlin bases for spaces of $k$-homogeneous polynomial solutions of the Hodge-de Rham system in $ ^m$ taking values in $s$-vectors, using Gegenbauer polynomials and an inductive construction. The key contribution is an algorithm to compute orthogonal bases for generalized Moisil-Théodoresco systems via these GT bases, valid for both real and complex Clifford algebras.

ABSTRACT

The main aim of this paper is to construct explicitly orthogonal bases for the spaces of k-homogeneous polynomial solutions of the Hodge-de Rham system in the Euclidean space R^m which take values in the space of s-vectors. Actually, we describe even the so-called Gelfand-Tsetlin bases for such spaces in terms of Gegenbauer polynomials. As an application, we obtain an algorithm how to compute an orthogonal basis of the space of homogeneous solutions of a generalized Moisil-Theodoresco system in R^m.

Motivation & Objective

• To explicitly construct orthogonal Gelfand-Tsetlin (GT) bases for the spaces $ H^s_k( ^m)$ of $k$-homogeneous polynomial solutions of the Hodge-de Rham system in $ ^m$ with values in $s$-vectors.
• To extend the Cauchy-Kovalevskaya (CK) method to the $H$-action on Clifford-algebra-valued polynomials for constructing GT bases in higher dimensions.
• To provide a dimension-by-dimension inductive algorithm for expressing GT bases in terms of Gegenbauer polynomials.
• To derive an algorithm for computing orthogonal bases of the space $ M^S_k( ^m)$ of homogeneous solutions of generalized Moisil-Théodoresco (GMT) systems using the constructed GT bases.
• To ensure the constructed bases are orthogonal with respect to invariant inner products, including the $L^2$ and Fischer inner products.

Proposed method

• Adapts the Cauchy-Kovalevskaya (CK) method to the $H$-action on $ ^m$-valued Clifford algebra polynomials to recursively construct GT bases for $ H^s_k( ^m)$.
• Employs an inductive construction on dimension $m$, expressing GT bases in terms of Gegenbauer polynomials via recursive relations.
• Uses the Fischer decomposition for the $H$-action to decompose the space $ M^S_k( ^m)$ into irreducible components involving $ H^s_k( ^m)$ and $ H^s_{k-1}( ^m)$.
• Applies the operator $(k-1+m-s)(xullet) - (k-1+s)(xullet)$ to GT bases of $ H^s_{k-1}( ^m)$ to generate parts of the basis for $ M^S_k( ^m)$.
• Handles the real Clifford algebra $ R_{0,m}$ by replacing complex conjugate pairs in the complex basis with real and imaginary parts to form a real GT basis.
• Validates the construction using symbolic computation in Maple with the Clifford package, enabling explicit computation of bases in low dimensions.

Experimental results

Research questions

• RQ1 How can orthogonal Gelfand-Tsetlin bases be explicitly constructed for the spaces $ H^s_k( ^m)$ of $k$-homogeneous solutions of the Hodge-de Rham system in $ ^m$?
• RQ2 What is the inductive structure of these GT bases in terms of Gegenbauer polynomials as the dimension $m$ increases?
• RQ3 How can the GT bases for $ H^s_k( ^m)$ be used to construct orthogonal bases for generalized Moisil-Théodoresco systems in $ ^m$?
• RQ4 What is the relationship between the $H$-action on $ M^S_k( ^m)$ and the irreducible components $ H^s_k( ^m)$ and $ H^s_{k-1}( ^m)$?
• RQ5 How can complex GT bases be transformed into real GT bases for the real Clifford algebra $ R_{0,m}$?

Key findings

• The paper provides an explicit inductive algorithm to construct orthogonal Gelfand-Tsetlin bases for $ H^s_k( ^m)$ in terms of Gegenbauer polynomials, valid for all $m$, $k$, and $s$.
• The constructed GT bases are orthogonal with respect to any invariant inner product, including the $L^2$ and Fischer inner products.
• For the complex Clifford algebra $ C_m$, the GT bases consist of complex-valued polynomials, and the method allows explicit computation using Maple with the Clifford package.
• In the real Clifford algebra $ R_{0,m}$, the GT bases are formed by replacing complex conjugate pairs with real and imaginary parts, yielding a real orthogonal basis.
• The space $ M^S_k( ^m)$ of homogeneous solutions of a generalized Moisil-Théodoresco system admits an orthogonal basis constructed as the union of GT bases $ B^{s,m}_k$ for $s otin S$ and transformed bases from $ H^s_{k-1}( ^m)$ via the $H$-action operator.
• Explicit examples of GT bases are computed in dimensions 3 and 4 for small $k$, demonstrating the feasibility and structure of the construction.