Solve telling me all my solutions are zero

There are two issues. One is that you did not provide the correct code for generating the equations. Waste of time for anyone who, like myself, tried to work on this (an immediate conclusion being, this wasted MY time).

Here is correct code.

ybe[a_, b_, c_, d_, e_, f_, R_, S_, T_] := Sum[Subscript[R, d, e, \[Alpha], \[Beta]] Subscript[S, \[Alpha], f, a, \[Gamma]] Subscript[T, \[Beta], \[Gamma], b, c] - Subscript[T, e, f, \[Beta], \[Gamma]] Subscript[S, d, \[Gamma], \[Alpha], c] Subscript[R, \[Alpha], \[Beta], a, b], {\[Alpha], 0, 1}, {\[Beta], 0, 1}, {\[Gamma], 0, 1}] Subscript[r[u_], k_, l_, i_, j_] := \[Delta][i + j, k + l] (\[Delta][i, j, k, l, 0] Subscript[u, 1] + \[Delta][i, j, k, l, 1] Subscript[u, 2] + \[Delta][i, k, 0] \[Delta][j, l, 1] Subscript[u, 3] + \[Delta][i, k, 1] \[Delta][j, l, 0] Subscript[u, 4] + \[Delta][i, l, 1] \[Delta][j, k, 0] Subscript[u, 5] + \[Delta][i, l, 0] \[Delta][j, k, 1] Subscript[u, 6]) /. \[Delta] -> KroneckerDelta exprs = DeleteCases[ Flatten[Table[ ybe[a, b, c, d, e, f, r[u], r[v], r[w]], {a, 0, 1}, {b, 0, 1}, {c, 0, 1}, {d, 0, 1}, {e, 0, 1}, {f, 0, 1}]], 0]; 

Exercise: What was the necessary change that corrects this?

The second issue is less obvious. All solutions other than the origin are nongeneric in that they force equations on some "constants". To get an idea of what these are one can enlarge the variable list e.g. by throwing the v variables into it.

vars = Join[Table[Subscript[v, i], {i, 6}], Table[Subscript[u, i], {i, 6}]]; solns = Solve[exprs == 0, vars]; 

This will give some idea of what happens in the solution set.

Length[solns] (* Out[199]= 36 *) solns[[1 ;; 4]] (* Out[203]= {{Subscript[u, 1] -> 0, Subscript[u, 2] -> 0, Subscript[u, 3] -> 0, Subscript[u, 4] -> 0, Subscript[u, 5] -> 0, Subscript[u, 6] -> 0}, {Subscript[u, 5] -> 0, Subscript[u, 6] -> 0, Subscript[v, 3] -> (Subscript[u, 3] Subscript[v, 1])/Subscript[u, 1], Subscript[v, 4] -> (Subscript[u, 4] Subscript[v, 2])/Subscript[ u, 2], Subscript[v, 5] -> 0, Subscript[v, 6] -> 0}, {Subscript[v, 1] -> 0, Subscript[v, 2] -> 0, Subscript[v, 3] -> 0, Subscript[v, 4] -> 0, Subscript[v, 5] -> 0, Subscript[v, 6] -> 0}, {Subscript[u, 1] -> 0, Subscript[u, 3] -> 0, Subscript[u, 5] -> 0, Subscript[u, 6] -> 0, Subscript[v, 4] -> (Subscript[u, 4] Subscript[v, 2])/Subscript[u, 2], Subscript[v, 5] -> 0, Subscript[v, 6] -> 0}} *) 

Some are uglier than those. Here is one such.

(* {Subscript[u, 1] -> (1/( 2 Subscript[v, 5] Subscript[w, 1]))(Subscript[u, 4] Subscript[v, 5] Subscript[w, 3] + ( Subscript[u, 4] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2])/ Subscript[w, 4] - ( Subscript[u, 4] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])/ Subscript[w, 4] - 1/(Subscript[u, 5] Subscript[w, 4])(\[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])^2))), Subscript[u, 2] -> (1/( 2 Subscript[v, 5] Subscript[w, 2]))((Subscript[u, 3] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2])/Subscript[w, 3] + Subscript[u, 3] Subscript[v, 5] Subscript[w, 4] - ( Subscript[u, 3] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])/ Subscript[w, 3] - 1/(Subscript[u, 4] Subscript[u, 5] Subscript[w, 3])Subscript[u, 3] \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])^2)), Subscript[u, 6] -> 0, Subscript[v, 1] -> (1/(2 \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) Subscript[w, 4] Subscript[w, 5]))(Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] - Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] - Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6] - \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])^2)), Subscript[v, 2] -> (1/( 2 Subscript[u, 4] Subscript[w, 3]))((Subscript[u, 3] Subscript[u, 4] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2])/(Subscript[u, 5] Subscript[w, 5]) - ( Subscript[u, 3] Subscript[u, 4] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4])/(Subscript[u, 5] Subscript[w, 5]) - ( Subscript[u, 3] Subscript[u, 4] Subscript[v, 5] Subscript[w, 6])/ Subscript[u, 5] - 1/(\!\( \*SubsuperscriptBox[\(u\), \(5\), \(2\)]\ \*SubscriptBox[\(w\), \(5\)]\))Subscript[u, 3] \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])^2)), Subscript[v, 3] -> (1/( 2 Subscript[u, 5] Subscript[w, 2]))((Subscript[u, 3] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2])/Subscript[w, 5] - ( Subscript[u, 3] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4])/ Subscript[w, 5] + Subscript[u, 3] Subscript[v, 5] Subscript[w, 6] - 1/(Subscript[u, 4] Subscript[u, 5] Subscript[w, 5])Subscript[u, 3] \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])^2)), Subscript[v, 4] -> ((2 Subscript[u, 4] \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(w\), \(1\), \(2\)]\) \!\(\*SubsuperscriptBox[\(w\), \(2\), \(2\)]\))/(Subscript[u, 5] \!\(\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\)) - (4 Subscript[u, 4] \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 1] Subscript[w, 2] Subscript[w, 3] Subscript[w, 4])/( Subscript[u, 5] \!\(\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\)) + (2 Subscript[u, 4] \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(w\), \(3\), \(2\)]\) \!\(\*SubsuperscriptBox[\(w\), \(4\), \(2\)]\))/(Subscript[u, 5] \!\(\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\)) - (2 Subscript[u, 4] \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 1] Subscript[w, 2] Subscript[w, 6])/( Subscript[u, 5] Subscript[w, 5]) - (2 Subscript[u, 4] \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 6])/( Subscript[u, 5] Subscript[w, 5]) - 1/(\!\( \*SubsuperscriptBox[\(u\), \(5\), \(2\)]\ \*SubsuperscriptBox[\(w\), \(5\), \(2\)]\))2 Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[ w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[ w, 5] Subscript[w, 6])^2) + 1/(\!\( \*SubsuperscriptBox[\(u\), \(5\), \(2\)]\ \*SubsuperscriptBox[\(w\), \(5\), \(2\)]\))2 Subscript[v, 5] Subscript[w, 3] Subscript[w, 4] \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[ w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[ w, 5] Subscript[w, 6])^2))/((2 Subscript[v, 5] \!\(\*SubsuperscriptBox[\(w\), \(1\), \(2\)]\) Subscript[w, 2])/ Subscript[w, 5] - ( 2 Subscript[v, 5] Subscript[w, 1] Subscript[w, 3] Subscript[w, 4])/Subscript[w, 5] - 2 Subscript[v, 5] Subscript[w, 1] Subscript[w, 6] - 1/(Subscript[u, 4] Subscript[u, 5] Subscript[w, 5])2 Subscript[w, 1] \[Sqrt](-4 \!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) \!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) \!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3] Subscript[w, 4] Subscript[w, 5] Subscript[w, 6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[ w, 3] Subscript[w, 4] + Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[ w, 5] Subscript[w, 6])^2)), Subscript[v, 6] -> 0} *)