Tools for SU(n)

Functions to help you study SU(n)'s irreducible representations.

I have now converted this MATLAB code to Python! If you prefer Python, click the following link, and don't look back! I will no longer be updating the MATLAB version.

https://github.com/hellpig/SU-tools

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MATLAB files

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These powerful functions study irreps of SU(n>1)...

• suf.m - given an irrep, finds all weights and their individual dimensions

• suw.m - using Weyl dimension formula, finds total dimension of all weights corresponding to an irrep

These very similar functions study specific SU(n) groups with the help of the above two functions...

• su2.m - plots all weights of an SU(2) irrep

• su3.m - plots all weights of an SU(3) irrep

• su4.m - plots all weights of an SU(4) irrep

• su5.m - plots all weights of an SU(5) irrep

• su6.m - plots a 4-dimensional slice of the weights of an SU(6) irrep

These functions study hadrons using all the above functions...

• hadrons.m - creates flavor-state multiplets for hadrons!

• hadr.m - used by hadrons.m to make a figure

• had.m - used by hadrons.m to make an animation

The following function uses suw.m...

• suy.m - finds all irreps within tensor products of SU(n>1) irreps

Run these in MATLAB making sure that the current folder contains these files. Use the help command to display help text (for example, type "help su3"). Octave, a free imitator of MATLAB, works very poorly with some of these functions due to its crappy graphics.

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my goal

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Representations of SU(n) interest me since they apply very much to the type of physics that is simple enough for me to presently understand. SU(2) describes spin-x particles where x is any non-negative half-integer or integer. SU(3) describes color charge. There are many other physical uses for the SU(n) groups, and I have created general functions that help you study irreducible representations and their weights. I have also written some functions that draw and animate hadron multiplets. I am fascinated by the way that observed hadrons belong to the 4-dimensional diagrams (which is a series of 3-dimensional diagrams) created by the bizarre and abstract math of special unitary groups.

Basically, I intended to write some code that very quickly does all the very long calculations required to study these very interesting groups while giving hadron flavor-states special attention.

The big picture of my code is that you will use L to define an irrep. The length of L is r, which means that SU(r+1) is being studied. To find out what I mean by all these symbols and words, see my list of terminology below.

I have done my best to make the code readable and expandable! Please conform it to your own needs and/or wants!

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terminology and other info

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Important vocab that I use...

• SU(n) — Lie group of all special unitary n×n matrices (I usually assume n>1)

• irrep — irreducible matrix representation of SU(n)

• weight — an "eigenvalue" of the irrep's Cartan subalgebra

• alpha — simple roots of the Lie algebra for SU(n) that have been given the geometric interpretation as vectors so that weights can be plotted, where it is true that (1) all positive roots are sums of these, (2) the order of simple roots is chosen to be the standard choice that makes the Cartan matrix have zeros everywhere except the 3 central diagonals, and (3) the default basis used to write these vectors is the standard rectangular basis in n-1 dimensional space

• alpha basis — I sometimes use alpha1=[1,0,0,...], alpha2=[0,1,0,...], etc. as a basis instead of the default rectangular basis

• Dynkin coefficients of a weight — every weight has integer Dynkin coefficients that uniquely label the weight, where these coefficients are the weight vector in a basis obtained by multiplying the inverse Cartan matrix with the column vector of alphas

• highest weight — this is the only weight of an irrep that can form all other weights by subtracting alphas (any list of non-negative Dynkin coefficients is a valid highest weight)

• dimension — I sometimes refer to the number of spatial dimensions needed to draw something, and I even sometimes mean the dimension of the SU(n) manifold, but I usually am referring to the dimension of a weight space (i.e. the multiplicity of a weight) or the total dimension of the direct sum of all weight spaces (note that the total dimension of an n×n irrep is n-dimensional)

Some variables I will use consistently...

• r = rank = number of alphas of a group = the number of Dynkin coefficients per weight = the number of spatial dimensions a plot of weights will require = rank of SU(r+1)

• L = Lambda = Dynkin coefficients of highest weight as a row vector

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See readme.html for the rest of the description!

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