Representations of \(\text{SL}(2,\mathbb{C})\) and unimodal polynomials.
An integral polynomial \(a_ 0+a_ 1t+...+a_ nt^ n\) is called symmetric unimodal if \(a_{n-v}=a_ v\) and \(0<a_ 0\leq a_ 1\leq...\leq a_{[n/2]}\). These polynomials, which arise in various combinatorial problems, are here studied in a new (and natural) way by associating with them certain (finite-dimensional, complex) representations of the group \(G=\text{SL}(2,\mathbb{C})\). Such a representation is called homogeneous if the degrees of its irreducible constituents are either all odd or all even. There is a bijection from the set of all symmetric unimodal polynomials to the set of all homogeneous representations of \(G\), having the property that the image of the product of two polynomials is the tensor product of their images. The irreducible \(G\)-modules may be denoted by \(V_ 1,V_ 2,...\), where \(\dim V_ n=n\). The fact that the exterior power \(\Lambda^ rV_ n\) affords a homogeneous representation yields a new proof that the Gaussian polynomial \(\left[\begin{matrix} n\\ r\end{matrix} \right]\) is symmetric unimodal. More generally, if \(\alpha\) is a partition then the Schur module \(S^{\alpha}V_ n\) affords a homogeneous representation, whence the Gaussian polynomial \(\left[\begin{matrix} n\\ \alpha \end{matrix} \right]\) is, symmetric unimodal. Similar considerations (involving the Adams operations) lead the author to conjecture that \(\prod^{r}_{v=1}(1- t^{vn})/(1-t^ v)\) is symmetric unimodal if \(n\) is even or if \(n\) is odd and \(r\) sufficiently large. This is known to be true for \(n=2\) and the author proves it for \(n=4\), using analytic methods of A. Odlyzko and B. Richmond to estimate the coefficients. Another conjecture put forward is that \((1+t)(1+t^ 3)...(1+t^{2n-1})+t^ 2+t^{n^ 2-2}\) is symmetric unimodal for \(n\geq 27\). The author also uses the (explicitly known) \(p\)-modular representation theory of \(G\) to produce still further symmetric unimodal polynomials. In a note added in proof, the author adverts to his subsequent proof of the second conjecture above [J. Pure Appl. Algebra 38, 121-126 (1985; Zbl 0584.05005)] and to his unpublished proof of the first conjecture for \(3\leq n\leq 20\) and \(n=100\), 101.
scientific article; zbMATH DE number 4012887