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rpp2010-rev-young-diagrams - 38 SU(n multiplets and Young...

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38. SU(n) multiplets and Young diagrams 1 38. SU( n ) MULTIPLETS AND YOUNG DIAGRAMS Written by C.G. Wohl (LBNL). This note tells (1) how SU( n ) particle multiplets are identified or labeled, (2) how to find the number of particles in a multiplet from its label, (3) how to draw the Young diagram for a multiplet, and (4) how to use Young diagrams to determine the overall multiplet structure of a composite system, such as a 3-quark or a meson-baryon system. In much of the literature, the word “representation” is used where we use “multiplet,” and “tableau” is used where we use “diagram.” 38.1. Multiplet labels An SU( n ) multiplet is uniquely identified by a string of ( n 1) nonnegative integers: ( α, β, γ, . . . ). Any such set of integers specifies a multiplet. For an SU(2) multiplet such as an isospin multiplet, the single integer α is the number of steps from one end of the multiplet to the other ( i.e. , it is one fewer than the number of particles in the multiplet). In SU(3), the two integers α and β are the numbers of steps across the top and bottom levels of the multiplet diagram. Thus the labels for the SU(3) octet and decuplet 1 1 0 3 are (1,1) and (3,0). For larger n , the interpretation of the integers in terms of the geometry of the multiplets, which exist in an ( n 1)-dimensional space, is not so readily apparent.
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