# Space and spin symmetry in light baryons

In Particle Physics by Martin and Shaw in Ch.6 they show how the baryon supermultiplets are built from the assumption that the space and spin wave functions are symmetrical, so eg. $uud$ has spins up,up,down so that under exchange of the $u$ quarks the wavefunction doesn't change sign as the spins are the same, but if a $u$ and $d$ is exchanged the spin changes so I assume the space wavefunction must introduce another minus from the flavour change to compensate the spin asymmetry and make the whole wavefunction symmetric. However, if say the $uds$ baryon has spin up,up,down then exchanging $u$ and $d$ particles leaves the spin wavefunction unchanged but if the $u$ and $d$ are exchanged it's not symmetrical due to the flavour part. So the $uud$ case requires an asymmetrical flavour wavefunction but $uds$ needs to be symmetrical under $u$ and $d$ exchange but antisymmetrical under exchange of the $s$ with the $u$ or $d$. So do you need to construct a spacial wavefunction for each spin state to make the whole thing symmetric? I feel like i'm missing something, it seems weird to have flavour change be asymmetric only sometimes, if someone could tell me where my thinkings going wrong or maybe justify the spacial wavefunction change I'd really appreciate it.

Chapter 6 of Martin & Shaw does seem to say that the $aa$ quark pair in an $aab$ baryon must be in a spin-1 state, e.g. that the spin-up proton wave function is $u{\uparrow}\;u{\uparrow}\;d{\downarrow}$. You are right to be confused by this, since the actual proton flavour-spin wave function is $$\frac{1}{\sqrt{18}}(2\;u{\uparrow}~u{\uparrow}~d{\downarrow} - u{\uparrow}~u{\downarrow}~d{\uparrow} -u{\downarrow}~u{\uparrow}~d{\uparrow} \\ \quad+ 2\;u{\uparrow} ~ d{\downarrow} ~ u{\uparrow} - u{\downarrow} ~ d{\uparrow} ~ u{\uparrow}-u{\uparrow} ~ d{\uparrow} ~ u{\downarrow} \\ \quad\;\;+ 2\;d{\downarrow} ~ u{\uparrow} ~ u{\uparrow} - d{\uparrow} ~u{\downarrow}~u{\uparrow} - d{\uparrow} ~ u{\uparrow} ~ u{\downarrow} ).$$ See, for example, the answers to Proton spin/flavor wavefunction or page 222 of this handout by Prof. Mark Thomson.