The action (Sp(4)xU(4)):M4,4 (C)
Chal Benson and Gail Ratcliff
This Maple worksheet provides details concerning an example presented in section 4.6.4 of our paper Spaces of Bounded Spherical Functions on Heisenberg Groups: Part I . At issue is the (multiplicity free) action of the compact group Sp(4)xU(4) on the space V=M4,4 (C) of 4x4 complex matrices via (k1, k2). Z = k1 Z k2^t . We will show that this action is well-behaved as defined in Section 2 of the paper.
restart: with(linalg):
Functions dot and sym below implement the inner product of 4x4 matrices and the symplectic product of vectors in C^4. In the inner product complex conjugation should be applied to the matrix entries of the second input. But for our purposes it will suffice to restrict attention to matrices all of whose entries are real.
dot:=(a,b)->sum(sum(a['u','v']*b['u','v'],'u'=1..4),'v'=1..4);
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sym:=(a,b)->sum(a['i']*b['i'+2]-a['i'+2]*b['i'],'i'=1..2);
Zio2JEkiYUc2IkkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3VtR0YlNiQsJiomJkYkNiMuSSJpR0YlIiIiJkYmNiMsJkYxRjMiIiNGM0YzRjMqJiZGJEY1RjMmRiZGMEYzISIiL0YxO0YzRjdGJUYlRiU=
So for example......
z:=matrix(4,4): dot(z,z); sym(col(z,1),col(z,2));
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This multiplicity free action has rank 6. Fundamental highest weights and highest weight vectors were given in a paper by Howe and Umeda. We implement these below as A1,...,A6 (highest weights) and h1,...h6 (highest weight vectors).
h1:=z->z[1,1];h1(z);
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Jkkiekc2IjYkIiIiRiY=
A1:=[[1,0],[1,0,0,0] ];
NyQ3JCIiIiIiITcmRiRGJUYlRiU=
h2:=z->z[1,1]*z[2,2]-z[1,2]*z[2,1];h2(z);
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A2:=[[1,1],[1,1,0,0]] ;
NyQ3JCIiIkYkNyZGJEYkIiIhRiY=
h3:=z->det(delcols(delrows(z,3...3),4..4)); h3(z);
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A3:=[[1,0],[1,1,1,0]];
NyQ3JCIiIiIiITcmRiRGJEYkRiU=
h4:=z->z[1,1]*z[3,2]-z[1,2]*z[3,1]+z[2,1]*z[4,2]-z[2,2 ]*z[4,1];
h4(z);
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A4:=[[0,0],[1,1,0,0]];
NyQ3JCIiIUYkNyYiIiJGJkYkRiQ=
m1:=z->matrix([[z[1,1],0,z[1,2],z[1,3]],
[z[2,1],0,z[2,2],z[2,3]],
[0,z[1,1],z[1,2], z[1,3]],
[0,z[3,1],z[3,2],z[3,3]]]):
m2:=z->matrix([[z[1,1],0,z[1,2],z[1,3]],
[z[2,1],0,z[2,2],z[2,3]],
[0,z[2,1],z[2,2], z[2,3]],
[0,z[4,1],z[4,2],z[4,3]]]):
h5:=z->det(m1(z))+det(m2(z));
h5(z);
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A5:=[[1,1],[2,1,1,0]];
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h6:=z->det(z);h6(z);
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A6:=[[0,0],[1,1,1,1]];
NyQ3JCIiIUYkNyYiIiJGJkYmRiY=
Matrix X will be an arbitrary element of the Lie algebra sp(4,C)....
X:=matrix(4,4): X[1,4]:= X[2,3]: X[3,3]:=-X[1,1]: X[4,3]:=-X[1,2]: X[3,4]:=-X[2,1]: X[4,4]:=- X[2,2]: X[4,1]:= X[3,2]: evalm(X) ;
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Matrix Y will be an arbitrary element of gl(4,C)....
Y:=matrix(4,4);
PTYiNiQ7IiIiIiIlRiVFXFtsIQ==
The moment map takes V to the dual of sp(4) x gl(4). We implement this below as function mom .
mom:=z-> simplify(dot(evalm(X&*z),z) ) +simplify(dot(evalm(z&*transpose(Y)),z));
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Using numerical methods, we found the points in V which map to diagonals under the moment map. We code the results of this investigation as procedure spt below. We will show that this produces a spherical point which maps to the weight aA1+bA2+cA3+dA4+eA5+fA6.
spt:=proc(a,b,c,d,e,f)
local z11,z12,z13,z14,z21,z22,z23,z24,z31,z32,z33,z34,z41,z42,z43,z44,sp;
z11:=(a*(a+b+e)*(a+c+e)*(a+b+c+2*e)*(a+2*b+c+d+2*e)*(a+b+c+d+e)*(a+b+c+d+2*e+f))/
((a+c)*(a+e)*(a+b+c+e)*(a+b+d+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e));
z12:=(b*c*e*(a+b+c+d+e)*(a+2*b+c+d+2*e)*(b+c+e)*(b+c+d+e+f))/
((a+c)*(a+e)*(b+d)*(a+b+c+e)*(a+2*b+c+2*e)*(b+c+d+e));
z13:=(c*d*(a+b+c+2*e)*(a+c+e)*(b+c+e)*(b+d+e)*(c+e+f))/
((a+c)*(b+d)*(c+e)*(a+b+c+e)*(a+2*b+c+2*e)*(a+b+d+e));
z14:=(a*b*d*e*f*(a+b+e)*(b+d+e))/
((a+c)*(c+e)*(a+b+c+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e)*(b+c+d+e));
z21:=(c*e*(a+b+e)*(a+b+c+2*e)*(a+2*b+c+d+2*e)*(b+d+e)*(a+b+c+d+2*e+f))/
((a+c)*(a+e)*(b+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e)*(a+b+d+e));
z22:=(a*b*(a+2*b+c+d+2*e)*(a+c+e)*(b+c+e)*(b+d+e)*(b+c+d+e+f))/
((a+c)*(a+e)*(b+d)*(b+e)*(a+2*b+c+2*e)*(b+c+d+e));
z23:=(a*d*e*(a+b+c+2*e)*(a+b+c+d+e)*(b+c+e)*(c+e+f))/
((a+c)*(b+d)*(b+e)*(c+e)*(a+2*b+c+2*e)*(a+b+d+e));
z24:=(b*c*d*f*(a+b+e)*(a+b+c+d+e)*(a+c+e))/
((a+c)*(b+e)*(c+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e)*(b+c+d+e));
z31:=(b*c*d*e*(b+c+e)*(b+d+e)*(a+b+c+d+2*e+f))/
((a+c)*(a+e)*(a+b+c+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e)*(a+b+d+e));
z32:=(a*d*(a+b+e)*(a+b+c+2*e)*(a+c+e)*(b+d+e)*(b+c+d+e+f))/
((a+c)*(a+e)*(b+d)*(a+b+c+e)*(a+2*b+c+2*e)*(b+c+d+e));
z33:=(a*b*e*(a+b+c+d+e)*(a+2*b+c+d+2*e)*(a+b+e)*(c+e+f))/
((a+c)*(b+d)*(c+e)*(a+b+c+e)*(a+2*b+c+2*e)*(a+b+d+e));
z34:=(c*f*(a+b+c+2*e)*(a+b+c+d+e)*(a+2*b+c+d+2*e)*(a+c+e)*(b+c+e))/
((a+c)*(c+e)*(a+b+c+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e)*(b+c+d+e));
z41:=(a*b*d*(a+b+c+d+e)*(a+c+e)*(b+c+e)*(a+b+c+d+2*e+f))/
((a+c)*(a+e)*(b+e)*(a+2*b+c+2*e)*(a+b+d+e)*(a+b+c+d+2*e));
z42:=(c*d*e*(a+b+c+2*e)*(a+b+c+d+e)*(a+b+e)*(b+c+d+e+f))/
((a+c)*(a+e)*(b+d)*(b+e)*(a+2*b+c+2*e)*(b+c+d+e));
z43:=(b*c*(a+2*b+c+d+2*e)*(a+b+e)*(a+c+e)*(b+d+e)*(c+e+f))/
((a+c)*(b+d)*(b+e)*(c+e)*(a+2*b+c+2*e)*(a+b+d+e));
z44:=(a*e*f*(a+b+c+2*e)*(a+2*b+c+d+2*e)*(b+d+e)*(c+b+e)*(c+d+e))/
((a+c)*(b+e)*(c+e)*(a+2*b+c+2*e)*(a+b+c+d+2*e)*(b+c+d+e)*(c+d+e));
sp:=matrix([[z11,z12,z13,z14],[z21,z22,z23,z24],[z31,z32,z33,z34],[z41,z42,z43,z44]]);
sp:=map(x->sqrt(x),sp);
sp[1,2]:=-sp[1,2]; sp[1,3]:=-sp[1,3]; sp[1,4]:=-sp[1,4]; sp[2,1]:=-sp[2,1]; sp[2,2]:=-sp[2,2]; sp[2,3]:=-sp[2,3];
sp[3,3]:=-sp[3,3]; sp[4,2]:=-sp[4,2];
evalm(sp);
end:
Here now is our general spherical point:
z:=spt(a,b,c,d,e,f);
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
To help with simplification, we replace each square root in the matrix above with a single variable, t1....t26.
getallterms:=proc(sp)
local ntermsarray, dtermsarray, Trms, i, j;
ntermsarray:=map(x->{op(convert(numer(x^2), list))}, sp);
dtermsarray:=map(x->{op(convert(denom(x^2), list))}, sp);
Trms:={};
for i from 1 to 4 do
for j from 1 to 4 do
Trms:=Trms union ntermsarray[i,j] union dtermsarray[i,j]
od od;
convert(Trms, list)
end:
Terms:=getallterms(z); nops(%);
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
IiNF
rewrite := proc(q)
local num, den, numr, denr, j:
global Terms;
num:={op(convert(numer(q), list))};
den:={op(convert(denom(q), list))};
numr:=1; denr:=1;
for j from 1 to nops(Terms) do
if Terms[j] in num then numr:=numr*t[j] fi;
if Terms[j] in den then denr:=denr*t[j] fi
od;
numr/denr
end:
The matrix zn is the spherical point expressed in terms of the t's.
zn:=zip((x,y)->sign(x)*(rewrite(y^2)), z, z);
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
To convert back from "t-variables" to parameters (a,b,...) use this substitution......
subt:={seq(t[j]=sqrt(Terms[j]), j=1..nops(Terms))};
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
Now we can apply the moment map to our general spherical point to show that we indeed get aA1+bA2+cA3+dA4+eA5+fA6=[[a+b+c+e,b+e],[a+b+c+d+2*e+f,b+c+d+e+f,c+e+f,f]]. Here recall that our weights are:
A1;A2;A3;A4;A5;A6;
NyQ3JCIiIiIiITcmRiRGJUYlRiU=
NyQ3JCIiIkYkNyZGJEYkIiIhRiY=
NyQ3JCIiIiIiITcmRiRGJEYkRiU=
NyQ3JCIiIUYkNyYiIiJGJkYkRiQ=
NyQ3JCIiIkYkNyYiIiNGJEYkIiIh
NyQ3JCIiIUYkNyYiIiJGJkYmRiY=
We apply the moment map to zn, convert back to parameters (a,...,f) , simplify and collect terms...
mom(zn):
subs(subt,%):
simplify(%):
collect(%,[X[1,1],X[2,2],Y[1,1],Y[2,2],Y[3,3],Y[4,4]]);
LC4qJiwqSSJhRzYiIiIiSSJiR0YmRidJImNHRiZGJ0kiZUdGJkYnRicmSSJYR0YmNiRGJ0YnRidGJyomLCZGKEYnRipGJ0YnJkYsNiQiIiNGMkYnRicqJiwuRiVGJ0YoRidGKUYnSSJkR0YmRicqJkYyRidGKkYnRidJImZHRiZGJ0YnJkkiWUdGJkYtRidGJyomLCxGKEYnRilGJ0Y1RidGKkYnRjdGJ0YnJkY5RjFGJ0YnKiYsKEYpRidGKkYnRjdGJ0YnJkY5NiQiIiRGQUYnRicqJkY3RicmRjk2JCIiJUZFRidGJw==
This completes the justification that z:=spt(a,b,c,d,e,f) is indeed a (generalized) spherical point for the weight aA1+bA2+cA3+dA4+eA5+fA6.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next we evaluate the highest weight vectors h1,...,h6 at our general spherical point z. For this we work in terms of the t's and simplify.
w:=matrix(4,4):
h1(w);
Jkkid0c2IjYkIiIiRiY=
h1(zn);
h1n:=subs(subt,%);
KjwmSSJ0RzYiNiMiIiJGJyZGJDYjIiM3RicmRiQ2IyIjOEYnJkYkNiMiIz1GJyZGJDYjIiNBRicmRiQ2IyIjQ0YnJkYkNiMiI0VGJyZGJDYjIiIoISIiJkYkNiMiIilGPSZGJDYjIiM8Rj0mRiQ2IyIjPkY9JkYkNiMiIz9GPSZGJDYjIiNCRj0=
KjwpSSJhRzYiIyIiIiIiI0YnKSwoRiRGJ0kiYkdGJUYnSSJlR0YlRidGJkYnKSwoRiRGJ0kiY0dGJUYnRixGJ0YmRicpLCpGJEYnRitGJ0YvRicqJkYoRidGLEYnRidGJkYnKSwsRiRGJ0YrRidGL0YnSSJkR0YlRidGLEYnRiZGJyksLEYkRicqJkYoRidGK0YnRidGL0YnRjVGJ0YyRidGJkYnKSwuRiRGJ0YrRidGL0YnRjVGJ0YyRidJImZHRiVGJ0YmRicpLCZGJEYnRi9GJ0YmISIiKSwmRiRGJ0YsRidGJkY+KSwqRiRGJ0YrRidGL0YnRixGJ0YmRj4pLCpGJEYnRitGJ0Y1RidGLEYnRiZGPiksKkYkRidGOEYnRi9GJ0YyRidGJkY+KSwsRiRGJ0YrRidGL0YnRjVGJ0YyRidGJkY+
h2(w);
LCYqJiZJIndHNiI2JCIiIkYoRigmRiU2JCIiI0YrRihGKComJkYlNiRGKEYrRigmRiU2JEYrRihGKCEiIg==
factor(h2(zn));subs(subt,%);h2n:=factor(%);
LCQqSCZJInRHNiI2IyIjNyIiIiZGJTYjIiM9RikmRiU2IyIjQUYpKSZGJTYjIiNDIiIjRikmRiU2IyIjRUYpJkYlNiNGNEYpJkYlNiMiIzlGKSZGJTYjIiM6RikmRiU2IyIjREYpLCYqJikmRiU2I0YpRjRGKSkmRiU2IyIjOEY0RilGKSomKSZGJTYjIiIkRjRGKSkmRiU2IyIiJkY0RilGKUYpKSZGJTYjIiIoRjQhIiIpJkYlNiMiIilGNEZZJkYlNiMiIzxGWSZGJTYjIiM+RlkpJkYlNiMiIz9GNEZZJkYlNiMiI0JGWSZGJTYjIiIqRlkmRiU2IyIjNUZZJkYlNiMiI0BGWUZZ
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
h3(w);
LC4qKCZJIndHNiI2JCIiIkYoRigmRiU2JCIiI0YrRigmRiU2JCIiJSIiJEYoRigqKEYkRigmRiU2JEYrRi9GKCZGJTYkRi5GK0YoISIiKigmRiU2JEYrRihGKCZGJTYkRihGK0YoRixGKEY1KihGN0YoJkYlNiRGKEYvRihGM0YoRigqKCZGJTYkRi5GKEYoRjlGKEYxRihGKCooRj9GKEY8RihGKUYoRjU=
factor(h3(zn));subs(subt,%):h3n:=factor(%);
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
h4(w);
LCoqJiZJIndHNiI2JCIiIkYoRigmRiU2JCIiJCIiI0YoRigqJiZGJTYkRihGLEYoJkYlNiRGK0YoRighIiIqJiZGJTYkRixGKEYoJkYlNiQiIiVGLEYoRigqJiZGJTYkRixGLEYoJkYlNiRGOEYoRihGMg==
factor(h4(zn));subs(subt,%):h4n:=factor(%);
KkImSSJ0RzYiNiMiI0EiIiImRiQ2IyIjQ0YoJkYkNiMiI0VGKCZGJDYjIiIlRigmRiQ2IyIjOkYoJkYkNiMiI0RGKCwqKiwpJkYkNiNGKCIiI0YoKSZGJDYjIiM3Rj1GKCkmRiQ2IyIjOEY9RigpJkYkNiMiIz1GPUYoKSZGJDYjIiM1Rj1GKEYoKiwpJkYkNiNGPUY9RigpJkYkNiMiIiRGPUYoKSZGJDYjIiImRj1GKCkmRiQ2IyIjOUY9RihGSkYoRigqLEZSRihGVkYoRj5GKEZGRigpJkYkNiMiIzxGPUYoRigqLEY6RihGT0YoRkJGKEZaRihGaW5GKEYoRigpJkYkNiMiIihGPSEiIikmRiQ2IyIiKUY9RmJvRmluRmJvJkYkNiMiIz5GYm8pJkYkNiMiIz9GPUZibyZGJDYjIiNCRmJvJkYkNiMiIipGYm8mRiQ2IyIjQEZib0ZKRmJv
KjYpLCxJImJHNiIiIiJJImNHRiZGJ0kiZEdGJkYnSSJlR0YmRidJImZHRiZGJyNGJyIiI0YnKSwoRiVGJ0YpRidGKkYnRixGJylGKUYsRicpLC5JImFHRiZGJ0YlRidGKEYnRilGJyomRi1GJ0YqRidGJ0YrRidGLEYnKSwsRjNGJyomRi1GJ0YlRidGJ0YoRidGKUYnRjRGJ0YsRicpLCxGM0YnRiVGJ0YoRidGKUYnRipGJ0YsRicpLCpGJUYnRihGJ0YpRidGKkYnRiwhIiIpLCZGJUYnRilGJ0YsRjwpLCxGM0YnRiVGJ0YoRidGKUYnRjRGJ0YsRjwpLCpGM0YnRiVGJ0YpRidGKkYnRixGPA==
h5(w);
LDoqKCkmSSJ3RzYiNiQiIiJGKSIiI0YpJkYmNiRGKkYqRikmRiY2JCIiJEYvRikhIiIqKEYkRikmRiY2JEYqRi9GKSZGJjYkRi9GKkYpRikqKkYlRikmRiY2JEYvRilGKSZGJjYkRilGL0YpRitGKUYpKipGJUYpRjdGKSZGJjYkRilGKkYpRjJGKUYwKiomRiY2JEYqRilGKUYlRilGPEYpRi1GKUYpKipGP0YpRiVGKUY5RilGNEYpRjAqKkY/RilGJUYpRitGKSZGJjYkIiIlRi9GKUYwKipGP0YpRiVGKUYyRikmRiY2JEZFRipGKUYpKigpRj9GKkYpRjxGKUZDRilGKSooRkpGKUY5RilGR0YpRjAqKkY/RikmRiY2JEZFRilGKUY8RilGMkYpRjAqKkY/RilGTUYpRjlGKUYrRilGKQ==
factor(h5(zn));h5n:=factor(subs(subt,%));
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
h6(w);
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
factor(h6(zn));h6n:=factor(subs(subt,%));
KkAmSSJ0RzYiNiMiI0UiIiImRiQ2IyIjREYoJkYkNiMiIztGKCZGJDYjIiInRigsUio2KSZGJDYjRigiIiVGKCkmRiQ2IyIjNyIiI0YoKSZGJDYjIiM4RjxGKCkmRiQ2IyIjPUY8RigpJkYkNiMiI0FGPEYoKSZGJDYjIiNDRjdGKCkmRiQ2I0Y8RjxGKCkmRiQ2IyIjOUY8RigpJkYkNiMiIzpGPEYoKSZGJDYjIiImRjxGKEYoKjYpRjVGPEYoRjhGKClGPkY3RihGQUYoRkVGKEZJRihGTUYoRlBGKEZURigpJkYkNiMiIiRGPEYoRigqNkY0RihGOEYoRj1GKClGQkY3RihGRUYoKUZKRjxGKCkmRiQ2I0Y3RjxGKEZURihGWEYoRlBGKEYoKjZGZ25GKClGOUY3RihGaG5GKEZBRihGRUYoRl9vRihGYG9GKEZURihGTUYoRmluRihGKCo2RmduRihGOEYoRj1GKEZeb0YoKUZGRjdGKEZfb0YoRmluRihGYG9GKEZYRihGUEYoRigqNkZnbkYoRmRvRihGPUYoRkFGKEZmb0YoRl9vRihGaW5GKEZgb0YoRlhGKEZNRighIiIqNkZpbkYoKUZZRjdGKEY4RihGVEYoRkFGKEZJRihGTUYoRlBGKEZFRihGZ25GKEYoKjYpRmpuRjdGKEZYRihGOEYoRlRGKEZBRihGSUYoRk1GKEZQRihGRUYoRj1GKEYoKjZGaW5GKEZYRihGOEYoKUZVRjdGKEZeb0YoRl9vRihGZ25GKEZgb0YoRj1GKEZQRihGKCo2RmluRihGWEYoRmRvRihGXnBGKEZBRihGX29GKEZnbkYoRmBvRihGPUYoRk1GKEZobyo2RlxwRihGWEYoRjhGKEZURihGXm9GKEZfb0YoRmBvRihGRUYoRj1GKEZQRihGKCo2RmluRihGam9GKEZkb0YoRlRGKEZBRihGX29GKEZgb0YoRkVGKEZnbkYoRk1GKEYoKjZGTUYoRmluRihGYG9GKEZqb0YoKUZRRjdGKEZURihGRUYoRl9vRihGZ25GKEZBRihGKCo2KUZORjdGKEZccEYoRmBvRihGWEYoRlBGKEZURihGRUYoRl9vRihGOEYoRj1GKEYoKjZGTUYoRmluRihGYG9GKEZYRihGY3BGKEZecEYoRmduRihGPUYoRl9vRihGQUYoRmhvKjZGZXBGKEZpbkYoRmBvRihGWEYoRlBGKEZecEYoRmduRihGOEYoRj1GKEZfb0YoRigqNkZNRihGaW5GKClGYW9GN0YoRmpvRihGUEYoRlRGKEZnbkYoRjhGKEZBRihGRUYoRigqNkZNRihGXHBGKEZpcEYoRlhGKEZQRihGVEYoRjhGKEZBRihGRUYoRj1GKEYoKjZGZ25GKEZNRihGYG9GKEY9RihGY3BGKEZmb0YoRmluRihGWEYoRl9vRihGQUYoRmhvKjZGZ25GKEZlcEYoRmBvRihGPUYoRlBGKEZmb0YoRmluRihGOEYoRlhGKEZfb0YoRigqNkZnbkYoRk1GKEZgb0YoRmhuRihGY3BGKEZFRihGVEYoRl9vRihGaW5GKEZBRihGKCo2RjRGKEZlcEYoRmBvRihGPUYoRlBGKEZFRihGVEYoRl9vRihGWEYoRjhGKEYoKjZGNEYoRk1GKEZpcEYoRj1GKEZQRihGRUYoRjhGKEZURihGQUYoRlhGKEYoKjZGZ25GKEZNRihGaXBGKEZobkYoRlBGKEZFRihGaW5GKEY4RihGVEYoRkFGKEYoRigpJkYkNiMiIihGN0ZobykmRiQ2IyIiKUY8RmhvKSZGJDYjIiM8RjxGaG8pJkYkNiMiIz5GPEZobykmRiQ2IyIjP0Y3RmhvKSZGJDYjIiNCRjxGaG8pJkYkNiMiIipGPEZobykmRiQ2IyIjNUY8RmhvKSZGJDYjIiNARjxGaG8pJkYkNiMiIzZGPEZobw==
KiopSSJmRzYiIyIiIiIiI0YnKSwoSSJjR0YlRidJImVHRiVGJ0YkRidGJkYnKSwsSSJiR0YlRidGK0YnSSJkR0YlRidGLEYnRiRGJ0YmRicpLC5JImFHRiVGJ0YvRidGK0YnRjBGJyomRihGJ0YsRidGJ0YkRidGJkYn
These are the formulas given in section 4.10 of our paper. They show that for positive real parameters (a,b,c,d,e,f) each fundamental highest weight vector hj (z) takes a non-zero value at z=spt(a,b,c,d,e,f). It follows that z lies in the open Borel orbit in V and Corollary 3.4 in our paper implies that (for positive integer values of the parameters a,...,f) the spherical point z is well-adapted to the highest weight vector h1 a h2 b h3 c h4 d h5 e h6 f . But we can also demonstrate this via direct computation as follows...
The well-adapted property uses directional derivatives of the HWV's evaluated at the spherical points. We simplify using the t-substitution. The i,j'th entry of dkn is the i,j'th derivative of h_k evaluated at the general spherical point (expressed in terms of the t's).
d1n:=matrix(4,4,(i,j)->coeff(coeff(h1(evalm(zn+t*w)),t),w[i,j])):
matrix(4,4,(i,j)->coeff(coeff(h2(evalm(zn+t*w)),t),w[i,j])):
d2n:=map(simplify,subs(subt,%)):
s:='s':matrix(4,4,(i,j)->coeff(coeff(h3(evalm(zn+s*w)),s),w[i,j])):
d3n:=map(simplify,subs(subt,%)):
matrix(4,4,(i,j)->coeff(coeff(h4(evalm(zn+s*w)),s),w[i,j])):
d4n:=map(simplify,subs(subt,%)):
matrix(4,4,(i,j)->coeff(coeff(h5(evalm(zn+s*w)),s),w[i,j])):
d5n:=map(simplify,subs(subt,%)):
matrix(4,4,(i,j)->coeff(coeff(h6(evalm(zn+s*w)),s),w[i,j])):
d6n:=map(simplify,subs(subt,%)):
The well-adapted condition says that the sum of the following six terms is the spherical point z.
term1:=map(simplify,evalm(a/h1n*d1n)):
term2:=map(simplify,evalm(b/h2n*d2n)):
term3:=map(simplify,evalm(c/h3n*d3n)):
term4:=map(simplify,evalm(d/h4n*d4n)):
term5:=map(simplify,evalm(e/h5n*d5n)):
term6:=map(simplify,evalm(f/h6n*d6n)):
ans:=map(factor, evalm(term1+term2+term3+term4+term5+term6));
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
The following code checks that this ans does equal the spherical point z by comparing the squares and signs of each entry in ans and z .
for i from 1 to 4 do;
for j from 1 to 4 do;
simplify(ans[i,j]^2/z[i,j]^2);
print(%);
end do;
end do;
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
IiIi
evalm(map(sign,z)-map(sign,ans));
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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our results hold for all positive reals parameters a...f. To complete the verification that our action is well-behaved we must also consider non-generic spherical points for which one or more coefficient a...f is zero. Here we apply Lemma 3.5 from our paper. First we check condition (3) in the Lemma: limits of (generalized) spherical points exist if we take some variables to zero. Since the moment map is continuous, these limits are also (generalized) spherical points....
First, we generate all possible ways the variables can go to zero. We eliminate the empty set. There are 63 non-empty subsets of {a=0, b=0, c=0, d=0, e=0, f=0}.
with(combinat):
zs:=[a=0,b=0,c=0,d=0,e=0,f=0];
ch:=[seq(choose(%)[i],i=2..64)]:
:nops(%);
NygvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm
IiNq
The following code lists each possible setting of parameters a...f to zero and prints the limiting (generalized) spherical point. This verifies condition (3) in Lemma 3.5. In particular no singularities arise as we perfom these limits via setting parameters to zero in succession.
for i from 1 to nops(ch) do:
print(ch[i]);
zf:=evalm(z):
for j from 1 to nops(ch[i]) do:
zf:=subs(ch[i][j],evalm(zf)):
end do:
print(zf);
end do:
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SSN6Zkc2Ig==
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To complete the verification that our action is well-behaved it remains to show that condition (4) in Lemma 3.5 applies. For each subset of the parameters (a,...,f) we obtained above a spherical point for the weight with those parameters set equal to zero. We must also check that the fundamental highest weight vectors associated with the complementary parameters are non-zero at this spherical point. For example at the spherical point Z for a weight of the sort aA1+0A2+cA3+0A4+0A5+fA6 (i.e. with b=d=e=0) we require that each of h1 (Z), h3 (Z) and h6 (Z) be non-zero.
The code below generates the following output for each of the 63 non-empty subsets of {a=0, b=0, c=0, d=0, e=0, f=0}:
- A listing of the subset. These parameters are set to zero in succession to obtain a non-generic (generalized) spherical point as in the previous output.
- A list of the fundamental highest weight vectors (h1 ...h6 ) associated with the complementary parameters.
- A list of values for these hj 's at the spherical point.
The output shows that in all cases each fundamental highest weight vector for a complementary parameter takes a non-zero value at the limiting spherical point in question. Thus condition (4) from Lemma 3.5 does hold here, completing our analysis for this example.
[1,2,3,4,5,6]: ch:=[seq(choose(%)[i],i=2..64)]:
h:=[h1,h2,h3,h4,h5,h6]: hn:=[h1n,h2n,h3n,h4n,h5n,h6n]:
zs:=[a=0,b=0,c=0,d=0,e=0,f=0]:
for i from 1 to nops(ch) do:
s:=map(x->zs[x],ch[i]):
print(s);
zc:=[op({1,2,3,4,5,6}minus{op(ch[i])})]:
hs:=map(x->hn[x],zc):
for j from 1 to nops(s) do:
hs:=subs(s[j],hs):
end do:
print(map(x->h[x],zc));
print(hs);
end do:
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