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Minuscule representation
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<p>In mathematical <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, a minuscule representation of a <a href="/facts/Semisimple_Lie_algebra/m8jA3Bcf">semisimple Lie algebra</a> or group is an <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible representation</a> such that the <a href="/facts/Weyl_group/N9QG4Wyn">Weyl group</a> acts transitively on the weights. Some authors exclude the <a href="/facts/Trivial_representation/2o9Fvjuj">trivial representation</a>. A quasi-minuscule representation (also called a basic representation) is an irreducible representation such that all non-zero weights are in the same orbit under the Weyl group; each simple Lie algebra has a unique quasi-minuscule representation that is not minuscule, and the multiplicity of the zero weight is the number of short nodes of the Dynkin diagram. (The highest weight of that quasi-minuscule representation is the highest short root, which in the simply-laced case is also the highest long root, making the quasi-minuscule representation be the <a href="/facts/Adjoint_representation/bWG4c2su">adjoint representation</a>.) </p><p>The minuscule representations are indexed by the <a href="/facts/Weight_lattice/7N8q5sw0">weight lattice</a> modulo the <a href="/facts/Root_lattice/TFaEdwMU">root lattice</a>, or equivalently by irreducible representations of the center of the simply connected <a href="/facts/Compact_group/Itl2HJaR">compact group</a>. For the simple Lie algebras, the dimensions of the minuscule representations are given as follows. </p> <ul><li>A<i>n</i> (<i>n</i>+1<i>k</i>) for 0 ≤ <i>k</i> ≤ <i>n</i> (exterior powers of vector representation). Quasi-minuscule: <i>n</i>2+2<i>n</i> (adjoint)</li> <li>B<i>n</i> 1 (trivial), 2<i>n</i> (spin). Quasi-minuscule: 2<i>n</i>+1 (vector)</li> <li>C<i>n</i> 1 (trivial), 2<i>n</i> (vector). Quasi-minuscule: 2<i>n</i>2–<i>n</i>–1 if <i>n</i>>1</li> <li>D<i>n</i> 1 (trivial), 2<i>n</i> (vector), 2<i>n</i>−1 (half spin), 2<i>n</i>−1 (half spin). Quasi-minuscule: 2<i>n</i>2–<i>n</i> (adjoint)</li> <li>E6 1, 27, 27. Quasi-minuscule: 78 (adjoint)</li> <li>E7 1, 56. Quasi-minuscule: 133 (adjoint)</li> <li>E8 1. Quasi-minuscule: 248 (adjoint)</li> <li>F4 1. Quasi-minuscule: 26</li> <li>G2 1. Quasi-minuscule: 7</li></ul> <ul><li>Seshadri, C. S. (1978), "Geometry of G/P. I. Theory of standard monomials for minuscule representations", <i>C. P. Ramanujam—a tribute</i>, Tata Inst. Fund. Res. Studies in Math., vol. 8, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 207–239</li></ul>