Schur Weyl Duality
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High Quality Content by WIKIPEDIA articles! Schur Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space mathbb{C}^notimesmathbb{C}^notimescdotsotimesmathbb{C}^n with k factors. The symmetric group Sk on k letters acts on this space by permuting the factors, sigma(v_1otimes v_2otimescdotsotimes v_k) = v_{sigma^{-1}(1)}otimes v_{sigma^{-1}(2)}otimescdotsotimes v_{sigma^{-1}(k)}. The general linear group GLn of invertible n×n matrices acts on it by the simultaneous matrix multiplication, g(v_1oti...
High Quality Content by WIKIPEDIA articles! Schur Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space mathbb{C}^notimesmathbb{C}^notimescdotsotimesmathbb{C}^n with k factors. The symmetric group Sk on k letters acts on this space by permuting the factors, sigma(v_1otimes v_2otimescdotsotimes v_k) = v_{sigma^{-1}(1)}otimes v_{sigma^{-1}(2)}otimescdotsotimes v_{sigma^{-1}(k)}. The general linear group GLn of invertible n×n matrices acts on it by the simultaneous matrix multiplication, g(v_1otimes v_2otimescdotsotimes v_k) = gv_1otimes gv_2otimescdotsotimes gv_k, quad gin GL_n.
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