The action (Sp(6)xU(3)):M6,3(C)

Chal Benson and Gail Ratcliff

This Maple worksheet provides details concerning an example presented in section 4.6.2 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(6)xU(3) on the space V=M6,3(C) of 6x3 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 6x3 matrices and the symplectic product of vectors in C^6. 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..6),'v'=1..3);

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sym:=(a,b)->sum(a['i']*b['i'+3]-a['i'+3]*b['i'],'i'=1..3);

Zio2JEkiYUc2IkkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3VtR0YlNiQsJiomJkYkNiMuSSJpR0YlIiIiJkYmNiMsJkYxRjMiIiRGM0YzRjMqJiZGJEY1RjMmRiZGMEYzISIiL0YxO0YzRjdGJUYlRiU=

So for example...

z:=matrix(6,3): 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=

h2:=z->z[1,1]*z[2,2]-z[1,2]*z[2,1];h2(z);

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LCYqJiZJInpHNiI2JCIiIkYoRigmRiU2JCIiI0YrRihGKComJkYlNiRGKEYrRigmRiU2JEYrRihGKCEiIg==

h3:=z->det(delrows(z,4..6));h3(z);

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A1:=[[1,0,0],[1,0,0]];

NyQ3JSIiIiIiIUYlRiM=

A2:=[[1,1,0],[1,1,0]] ;

NyQ3JSIiIkYkIiIhRiM=

A3:=[[1,1,1],[1,1,1]];

NyQ3JSIiIkYkRiRGIw==

h4:=z->sym(col(z,1),col(z,2));h4(z);

Zio2I0kiekc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3ltR0YlNiQtX0knbGluYWxnRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlSSRjb2xHRiU2JEYkIiIiLUYtNiRGJCIiI0YlRiVGJQ==

A4:=[[0,0,0], [1,1,0]] ;

NyQ3JSIiIUYkRiQ3JSIiIkYmRiQ=

h5:=z->expand(z[1,1]*sym(col(z,2),col(z,3))-z[1,2]*sym(col(z,1),col(z,3))+z[1,3]*sym(col(z,1),col(z,2)));h5(z);

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A5:=[[1,0,0], [1,1,1]];

NyQ3JSIiIiIiIUYlNyVGJEYkRiQ=

h6:=z->expand((z[1,1]*z[2,3]-z[2,1]*z[1,3])*sym(col(z,1),col(z,2))- (z[1,1]*z[2,2]-z[2,1]*z[1,2])*sym(col(z,1),col(z,3))); h6(z);

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A6:=[[1,1,0], [2,1,1]];

NyQ3JSIiIkYkIiIhNyUiIiNGJEYk

A:=[A1,A2,A3,A4,A5,A6];

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Matrix X will be an arbitrary element of the Lie algebra sp(6,C)....

X:=matrix(6,6): X[1,5]:=X[2,4]: X[1,6]:=X[3,4]: X[4,4]:=-X[1,1]: X[5,4]:=-X[1,2]: X[6,4]:=-X[1,3]: X[2,6]:=X[3,5]: X[4,5]:=-X[2,1]: X[5,5]:=-X[2,2]: X[6,5]:=-X[2,3]: X[1,6]:=X[3,4]: X[4,6]:=-X[3,1]: X[5,6]:=-X[3,2]: X[6,6]:=-X[3,3]: X[4,4]:=-X[1,1]: X[4,2]:=X[5,1]: X[4,3]:=X[6,1]: X[5,3]:=X[6,2]: evalm(X);

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Matrix Y will be an arbitrary element of gl(3,C)....

Y:=matrix(3,3):

The moment map takes V to the dual of sp(6) x gl(3). 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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To illustrate the use of function mom we give below spherical points for each of A1, A2, A3, A4, A5, A6....

z:=matrix(6,3,0): z[1,1]:=1: evalm(z); mom(z); A1;

LCYmSSJYRzYiNiQiIiJGJ0YnJkkiWUdGJUYmRic=

NyQ3JSIiIiIiIUYlRiM=

z:=matrix(6,3,0): z[1,1]:=1: z[2,2]:=1: evalm(z); mom(z); A2;

LComSSJYRzYiNiQiIiJGJ0YnJkYkNiQiIiNGKkYnJkkiWUdGJUYmRicmRixGKUYn

NyQ3JSIiIkYkIiIhRiM=

z:=matrix(6,3,0): z[1,1]:=1: z[2,2]:=1: z[3,3]:=1: evalm(z); mom(z); A3;

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NyQ3JSIiIkYkRiRGIw==

z:=matrix(6,3,0): z[1,1]:=1: z[4,2]:=1: evalm(z); mom(z); A4;

LCYmSSJZRzYiNiQiIiJGJ0YnJkYkNiQiIiNGKkYn

NyQ3JSIiIUYkRiQ3JSIiIkYmRiQ=

z:=matrix(6,3,0): z[1,1]:=1: z[2,2]:=1: z[5,3]:=1: evalm(z); mom(z); A5;

LComSSJYRzYiNiQiIiJGJ0YnJkkiWUdGJUYmRicmRik2JCIiI0YsRicmRik2JCIiJEYvRic=

NyQ3JSIiIiIiIUYlNyVGJEYkRiQ=

z:=matrix(6,3,0): z[1,1]:=sqrt(2): z[2,2]:=1: z[4,3]:=1: evalm(z); mom(z); A6;

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NyQ3JSIiIkYkIiIhNyUiIiNGJEYk

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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,Z21,Z22,Z23,Z31,Z32,Z33,Z41,Z42,Z43,Z51,Z52,Z53,Z61,Z62,Z63; Z11:=a*(a+e+f)*(a+b+f)*(a+b+e+d+f)*(a+b+2*c+e+2*f)*(a+2*b+2*c+d+e+2*f)*(c+e+f+a+b) /(a+e)/(a+f)/(a+b+2*c+e+f)/(a+b+d+f)/(a+2*b+2*c+e+2*f)/(e+f+a+b): Z12:=b*e*f*(a+2*b+d+2*c+e+2*f)*(b+2*c+e+f)*(a+b+d+e+f)*(a+b+c+e+f) /(a+2*b+2*c+e+2*f)/(a+b+2*c+e+f)/(a+e)/(a+f)/(a+b+e+f)/(b+d): Z13:=e*d*(2*c+2*f+e+a+b)*(b+d+f)*(c+e+f+a+b)*(2*c+e+f+b)*(e+f+a) /(a+2*b+2*c+2*f+e)/(a+e)/(b+d)/(a+b+2*c+e+f)/(a+b+d+f)/(e+f+a+b): Z21:=e*f*(e+2*c+2*f+a+2*b+d)*(e+2*c+2*f+a+b)*(a+b+f)*(b+d+f)*(c+f+b) /(2*c+2*f+e+a+2*b)/(e+a)/(f+a)/(b+f)/(a+b+d+f)/(2*c+f+b): Z22:=(2*c+2*f+e+a+2*b+d)*a*b*(b+d+f)*(e+2*c+f+b)*(e+f+a)*(c+f+b) /(2*c+2*f+e+a+2*b)/(a+e)/(b+d)/(b+f)/(a+f)/(2*c+f+b): Z23:=a*d*f*(e+2*c+2*f+a+b)*(e+f+2*c+b)*(e+f+a+b+d)*(c+f+b) /(e+2*c+2*f+a+2*b)/(b+d)/(a+e)/(b+f)/(a+b+d+f)/(f+2*c+b): Z31:=a*c*(2*c+e+2*f+a+2*b+d)*e*(2*c+2*f+e+a+b) /(e+2*c+f+a+b)/(f+2*c+b)/(e+f+a+b)/(f+a+b+d)/(f+a) *b/(b+f)*d: Z32:=(e+2*c+2*f+a+2*b+d)*(e+2*c+f+b)*c*f*(f+a+b)*(e+f+a) /(f+a)*(f+b)/(e+2*c+f+a+b)/(f+2*c+b)/(e+f+a+b) /(b+f)/(b+f)*d/(b+d): Z33:=(e+2*c+2*f+a+b)*(e+2*c+f+b)*(e+f+a+b+d)*c*(f+a+b)*(f+b+d) /(e+2*c+f+a+b)/(f+2*c+b)/(e+f+a+b)/(f+a+b+d)/(f+b) *b/(b+d): Z41:=e*f*b*d*(e+f+2*c+b)*(b+d+f)*(e+f+c+a+b) /(e+2*c+f+a+b)/(2*c+2*f+e+a+2*b)/(a+e)/(a+f)/(a+b+d+f)/(e+f+a+b): Z42:=(e+2*c+2*f+a+b)*(c+e+f+a+b)*a*d*(a+b+f)*(e+f+a)*(b+d+f) /(2*c+f+e+a+b)/(2*c+2*f+e+a+2*b)/(a+e)/(e+f+a+b)/(b+d)/(a+f): Z43:=a*b*f*(e+2*c+2*f+a+2*b+d)*(e+f+c+a+b)*(e+f+a+b+d)*(a+b+f) /(2*c+f+e+a+b)/(2*c+2*f+e+a+2*b)/(a+e)/(b+d)/(e+f+a+b)/(a+b+d+f): Z51:=a*b*d*(e+f+2*c+b)*(f+c+b)*(e+f+a+b+d)*(e+f+a) /(2*c+2*f+e+a+2*b)/(f+2*c+b)/(a+e)/(a+f)/(b+f)/(a+b+d+f): Z52:=e*d*f*(2*c+2*f+e+a+b)*(c+f+b)*(e+f+a+b+d)*(a+b+f) /(a+e)/(a+f)/(b+d)/(b+f)/(2*c+e+2*f+a+2*b)/(f+2*c+b): Z53:=b*e*(a+b+f)*(a+e+f)*(b+c+f)*(b+d+f)*(a+2*b+2*c+d+e+2*f) /(a+e)/(b+d)/(b+f)/(a+b+d+f)/(b+2*c+f)/(a+2*b+2*c+e+2*f): Z61:=(e+2*c+f+b)*(e+f+a+b+d)*c*f*(e+f+a)*(f+a+b)*(f+b+d) /(e+2*c+f+a+b)/(f+2*c+b)/(e+f+a+b)/(f+a)/(f+a+b+d)/(b+f): Z62:=(e+2*c+2*f+a+b)*(e+f+a+b+d)*c*e /(e+2*c+f+a+b)/(e+f+a+b)/(f+2*c+b)/(f+a) *(b+d+f)*b/(b+f)/(b+d)*a: Z63:=a*(e+2*c+2*f+a+2*b+d)*c*e*f*(e+f+a) /(e+2*c+f+a+b)/(f+2*c+b)/(e+f+a+b)/(f+b)/(f+a+b+d) /(b+d)*d: matrix([[sqrt(Z11),-sqrt(Z12),sqrt(Z13)], [sqrt(Z21),sqrt(Z22),-sqrt(Z23)], [-sqrt(Z31),sqrt(Z32),sqrt(Z33)], [sqrt(Z41),sqrt(Z42),sqrt(Z43)], [-sqrt(Z51),sqrt(Z52),sqrt(Z53)], [-sqrt(Z61),-sqrt(Z62),sqrt(Z63)]]) end:

Below we apply procedure spt to 6 random integer inputs in the range 1..10. Applying the moment map yields a "diagonal functional" as required. Re-executing these lines will produce a new numerical example each time.

r:=rand(1..10):

z:=spt(r(),r(),r(),r(),r(),r());

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mom(z);

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Here is our general spherical point:

sp:=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....t24.

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 6 do for j from 1 to 3 do Trms:=Trms union ntermsarray[i,j] union dtermsarray[i,j] od od; convert(Trms, list) end:

Terms:=getallterms(sp); nops(Terms);

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IiND

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 zt is the spherical point expressed in terms of the t's.

zt:=zip((x,y)->sign(x)*(rewrite(y^2)), sp, sp);

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

To convert back from "t-variables" to parameters (a,b,...) use this substitution......

Sub:={seq(t[j]=sqrt(Terms[j]), j=1..nops(Terms))};

PDovJkkidEc2IjYjIiIiKiQpSSJhR0YmI0YoIiIjRigvJkYlNiNGLSokKUkiYkdGJkYsRigvJkYlNiMiIiQqJClJImNHRiZGLEYoLyZGJTYjIiIlKiQpSSJkR0YmRixGKC8mRiU2IyIiJiokKUkiZUdGJkYsRigvJkYlNiMiIicqJClJImZHRiZGLEYoLyZGJTYjIiIoKiQpLCZGK0YoRkhGKEYsRigvJkYlNiMiIikqJCksJkYrRihGT0YoRixGKC8mRiU2IyIiKiokKSwmRjNGKEZBRihGLEYoLyZGJTYjIiM1KiQpLCZGM0YoRk9GKEYsRigvJkYlNiMiIzYqJCksKEYrRihGM0YoRk9GKEYsRigvJkYlNiMiIzcqJCksKEYrRihGSEYoRk9GKEYsRigvJkYlNiMiIzgqJCksKEYzRihGQUYoRk9GKEYsRigvJkYlNiMiIzkqJCksKEY6RihGT0YoRjNGKEYsRigvJkYlNiMiIzoqJCksKComRi1GKEY6RihGKEZPRihGM0YoRixGKC8mRiU2IyIjOyokKSwqRitGKEYzRihGQUYoRk9GKEYsRigvJkYlNiMiIzwqJCksKkYzRihGaXFGKEZIRihGT0YoRixGKC8mRiU2IyIjPSokKSwqRkhGKEZPRihGK0YoRjNGKEYsRigvJkYlNiMiIz4qJCksLEYrRihGM0YoRmlxRihGSEYoRk9GKEYsRigvJkYlNiMiIz8qJCksLEYrRihGM0YoRmlxRihGSEYoKiZGLUYoRk9GKEYoRixGKC8mRiU2IyIjQCokKSwsRitGKEYzRihGSEYoRkFGKEZPRihGLEYoLyZGJTYjIiNBKiQpLCxGK0YoKiZGLUYoRjNGKEYoRmlxRihGSEYoRl10RihGLEYoLyZGJTYjIiNCKiQpLCxGOkYoRkhGKEZPRihGK0YoRjNGKEYsRigvJkYlNiMiI0MqJCksLkYrRihGXHVGKEZpcUYoRkFGKEZIRihGXXRGKEYsRig=

We apply the moment map to zt, convert back to parameters (a,...,f) , simplify and collect terms...

mom(zt): subs(Sub,%): simplify(%): collect(%,[X[1,1],X[2,2],X[3,3],Y[1,1],Y[2,2],Y[3,3]]);

LC4qJiwsSSJjRzYiIiIiSSJlR0YmRidJImZHRiZGJ0kiYUdGJkYnSSJiR0YmRidGJyZJIlhHRiY2JEYnRidGJ0YnKiYsKEYlRidGKUYnRitGJ0YnJkYtNiQiIiNGM0YnRicqJkYlRicmRi02JCIiJEY3RidGJyomLC5GKEYnKiZGM0YnRilGJ0YnRipGJ0YrRidJImRHRiZGJ0YlRidGJyZJIllHRiZGLkYnRicqJiwsRilGJ0Y7RidGKEYnRiVGJ0YrRidGJyZGPUYyRidGJyomLChGJUYnRihGJ0YpRidGJyZGPUY2RidGJw==

So this is a "diagonal functional" and the weight for aA1+bA2+cA3+dA4+eA5+fA6 is:

evalm(a*A1+b*A2+c*A3+d*A4+e*A5+f*A6);

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

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 sp. For this we work in terms of the t's and simplify.

h1(zt):h1n:=subs(Sub,%);

KjwpSSJhRzYiIyIiIiIiI0YnKSwoRiRGJ0kiYkdGJUYnSSJmR0YlRidGJkYnKSwoRiRGJ0kiZUdGJUYnRixGJ0YmRicpLCxGJEYnRitGJyomRihGJ0kiY0dGJUYnRidGL0YnKiZGKEYnRixGJ0YnRiZGJyksLEYkRidGK0YnRi9GJ0kiZEdGJUYnRixGJ0YmRicpLCxGM0YnRi9GJ0YsRidGJEYnRitGJ0YmRicpLC5GJEYnKiZGKEYnRitGJ0YnRjJGJ0Y3RidGL0YnRjRGJ0YmRicpLCZGJEYnRi9GJ0YmISIiKSwmRiRGJ0YsRidGJkY/KSwqRiRGJ0YrRidGN0YnRixGJ0YmRj8pLCpGL0YnRixGJ0YkRidGK0YnRiZGPyksLEYkRidGK0YnRjJGJ0YvRidGLEYnRiZGPyksLEYkRidGPEYnRjJGJ0YvRidGNEYnRiZGPw==

h2(zt):factor(%):subs(Sub,%):h2n:=factor(%);

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

h3(zt):factor(%):subs(Sub,%):h3n:=factor(%);

KjQpSSJjRzYiIyIiIiIiI0YnLCpJImJHRiVGJyomRihGJ0YkRidGJ0kiZUdGJUYnSSJmR0YlRidGJyksKEYkRidGLUYnRipGJ0YmRicsLkkiYUdGJUYnKiZGKEYnRipGJ0YnRitGJ0kiZEdGJUYnRixGJyomRihGJ0YtRidGJ0YnKSwsRiRGJ0YsRidGLUYnRjFGJ0YqRidGJkYnLCxGMUYnRipGJ0YrRidGLEYnRjRGJ0YnLChGK0YnRi1GJ0YqRichIiIsLEYxRidGKkYnRitGJ0YsRidGLUYnRjksLEYxRidGMkYnRitGJ0YsRidGNEYnRjk=

h4(zt):factor(%):subs(Sub,%):h4n:=factor(%);

Ki4pLChJImJHNiIiIiJJImRHRiZGJ0kiZkdGJkYnI0YnIiIjRicpRihGKkYnKSwuSSJhR0YmRicqJkYrRidGJUYnRicqJkYrRidJImNHRiZGJ0YnRihGJ0kiZUdGJkYnKiZGK0YnRilGJ0YnRipGJyksLEYvRidGJUYnRjNGJ0YoRidGKUYnRipGJyksJkYlRidGKEYnRiohIiIpLCpGL0YnRiVGJ0YoRidGKUYnRipGOQ==

factor(h5(zt)):subs(Sub,%):h5n:=factor(%);

KjgpLCZJImFHNiIiIiJJImVHRiZGJyNGJyIiIyEiIiksKkYoRidJImZHRiZGJ0YlRidJImJHRiZGJ0YpRispLCxGJUYnRi9GJyomRipGJ0kiY0dGJkYnRidGKEYnRi5GJ0YpRispLCxGJUYnKiZGKkYnRi9GJ0YnRjJGJ0YoRicqJkYqRidGLkYnRidGKUYrKUYoRilGJyksKkYvRidGMkYnRihGJ0YuRidGKUYnKSwuRiVGJ0Y2RidGMkYnSSJkR0YmRidGKEYnRjdGJ0YpRicpLCxGM0YnRihGJ0YuRidGJUYnRi9GJ0YpRicpLCxGJUYnRi9GJ0YoRidGPUYnRi5GJ0YpRicpLCxGJUYnRi9GJ0YyRidGKEYnRjdGJ0YpRicpLChGJUYnRihGJ0YuRidGKUYn

factor(h6(zt)):subs(Sub,%):h6n:=factor(%);

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These are the formulas given in section 4.8 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=sp=spt(a,b,c,d,e,f). It follows that sp 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 sp is well-adapted to the highest weight vector h1a h2b h3c h4d h5e h6f . 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 dk is the i,j'th derivative of hk evaluated at the spherical point sp.

u:=matrix(6,3):w:=matrix(6,3):

d1:=matrix(6,3,(i,j)->coeff(coeff(h1(evalm(zt+p*w)),p),w[i,j])):

matrix(6,3,(i,j)->coeff(coeff(h2(evalm(zt+p*w)),p),w[i,j])): d2:=map(factor,subs(Sub,%)):

matrix(6,3,(i,j)->coeff(coeff(h3(evalm(zt+p*w)),p),w[i,j])): d3:=subs(Sub,%):

matrix(6,3,(i,j)->coeff(coeff(h4(evalm(zt+p*w)),p),w[i,j])): d4:=map(factor,subs(Sub,%)):

matrix(6,3,(i,j)->coeff(coeff(h5(evalm(zt+p*w)),p),w[i,j])):map(factor,%): d5:=map(factor,subs(Sub,%)):

matrix(6,3,(i,j)->coeff(coeff(h6(evalm(zt+p*w)),p),w[i,j])):map(factor,%): d6:=subs(Sub,%):

The well-adapted condition says that the sum of the following six terms is the spherical point sp.

term1:=map(factor,evalm((a)/h1n*d1)):

term2:=map(factor,evalm(b/h2n*d2)):

term3:=map(factor,evalm(c/h3n*d3)):

term4:=map(simplify,evalm(d/h4n*d4)):

term5:=map(factor,evalm(e/h5n*d5)):

term6:=map(factor,evalm(f/h6n*d6)):

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 sp by comparing the squares and signs of each entry in ans and sp.

for i from 1 to 6 do; for j from 1 to 3 do; simplify(ans[i,j]^2/sp[i,j]^2); print(%); end do; end do;

IiIi

evalm(map(sign,sp)-map(sign,ans));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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(%);

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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(sp): 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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