Quantum alpha-determinants and q-deformed hypergeometric polynomials

The quantum $\alpha$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$ generated by the powers of the quantum $\alpha$-determinant. For such a cyclic module, there exists a collection of polynomials which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible $\mathcal{U}_q(\mathfrak{sl}_2)$-module, (ii) the cyclic module contains an irreducible submodule if the parameter is a root of the corresponding polynomial. These polynomials are given as a $q$-deformation of the hypergeometric polynomials. This is a quantum analogue of the result obtained in our previous work [K. Kimoto, S. Matsumoto and M. Wakayama, Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans. Amer. Math. Soc.].Comment: 10 page

Human cell line (HGC-27) derived from the metastatic lymph node of gastric cancer

A cell line (HGC-27) was established by culture of the metastatic lymph node from a gastric cancer patient diagnosed histologically as undifferentiated carcinoma. HGC-27 cells were polygonal or short spindle-shaped and adhered to glass surfaces as a monolayer. The cells were probably derived from gastric cancer cells, as their origin from mesenchymal tissues can be excluded morphologically and enzyme-histochemically. Enzyme activities were generally negative or low, except for adenosine triphosphatase, lactic dehydrogenase and leucine aminopeptidase. These scanty findings might reflect the undifferentiated character of the original tumor cells. The cloning efficiency was 5.3% in liquid medium and 1.0% in soft agar. The doubling time was about 17 hr. Chromosomal analysis revealed a mode of 109 and 110 chromosomes.</p