Given two irreducible representations \(V_\lambda\), \(V_\mu\) of \(\text{GL}_r(\mathbb{C})\), where \(\lambda= (\lambda_1\geq\cdots\geq \lambda_r\geq 0)\), \(\mu= (\mu_1\geq\cdots\geq \mu_r\geq 0)\), let \(c^\nu_{\lambda,\mu}\) denote the Littlewood-Richardson coefficient, namely, the coefficient with which \(V_\nu\) occurs in \(V_\lambda\otimes V_\mu\), and let \(d^\nu_{\lambda,\mu}\) denote the structure coefficient (the coefficient with which the Schubert class \(\sigma_\nu\) occurs in \(\sigma_\lambda\cdot\sigma_\mu\)) in \(H^\bullet(G_{r,n})\), where \(n\) is such that \(\lambda_1\), \(\mu_1\) are \(\leq n-r\), and \(G_{r,n}\) is the Grassmannian variety of \(r\)-dimensional subspaces of \(\mathbb{C}^n\). It is a well-known fact that \(c^\nu_{\lambda,\mu}= d^\nu_{\lambda,\mu}\).
In this article, the author gives a geometric proof of the (weaker) result \(d^\nu_{\lambda,\mu}\leq c^\nu_{\lambda,\mu}\) using tangent spaces to Schubert varieties; as a consequence, the author describes bases for \(\text{SL}_r(\mathbb{C})\)-invariants for the tensor products of irreducible \(\text{SL}_r(\mathbb{C})\)-modules. As the author observes, the tangent space approach adopted in this paper may provide the key for analogous results for other types. This article makes an important contribution to the study of flag varieties – their geometric and representation-theoretic aspects.