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On O m ×G L n Highest Weight Vectors

In: Symmetries in Science VIII

Author

Listed:
  • Helmer Aslaksen

    (National University of Singapore, Department of Mathematics)

  • Eng-Chye Tan

    (National University of Singapore, Department of Mathematics)

  • Chen-bo Zhu

    (National University of Singapore, Department of Mathematics)

Abstract

Let ℂm,n be the vector space of m×n complex matrices and P(ℂm,n) be the algebra of complex-valued polynomials on ℂm,n. Let GL m ×GL n act on P(ℂm,n) by pre-and post-multiplication as follows: $$\left( {{g_1},{g_2}} \right)f\left( x \right) = f\left( {g_1^{ - 1}x{g_2}} \right)$$ where x ∈ ℂm,n, (g 1,g 2) ∈ GL m ×GL n,f ∈ P(ℂm,n). We choose a system of coordinates on ℂm,n as follows: $$\left[ {\begin{array}{*{20}{c}}{{x_{11}}}{{x_{12}}} \ldots {{x_{1n}}} \\{{x_{21}}}{{x_{22}}} \ldots {{x_{2n}}} \\\vdots \vdots \cdots \vdots \\{{x_{m1}}}{{x_{m2}}} \cdots {{x_{mn}}}\end{array}} \right]$$

Suggested Citation

  • Helmer Aslaksen & Eng-Chye Tan & Chen-bo Zhu, 1995. "On O m ×G L n Highest Weight Vectors," Springer Books, in: Bruno Gruber (ed.), Symmetries in Science VIII, pages 1-11, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-1915-7_1
    DOI: 10.1007/978-1-4615-1915-7_1
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