Non-Abelian 3-node quivers

J. Manschot, B. Pioline, A.Sen, Feb 2013. Updated for CoulombHiggs 2.0, Sep 2013; Updated for CoulombHiggs 6.0, Dec 2020

SetDirectory[NotebookDirectory[]];
<<CoulombHiggs`

Choosing the DSZ matrix, dimension vector and FI terms

Evaluate the cell corresponding to the dimension vector, and proceed to section “Evaluate the Poincare polynomial”.
The result of the computation is recorded below as OmR[Nvec], with corresponding list of  H-factors as soH[Nvec]

Rank (1,1,1)

aa=3;bb=2;cc=3;Nvec={1,1,1};Cvec={1,-8/10,-2/10};
k=kk(aa+bb-cc);
soH[{1,1,1}]={CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,1},y_]->-((2 (y^2))/(((-1+(y^2)))^2))};
OmR[{1,1,1}]=OmS[{1,1,1}]+((1+(-1)^k)/2(y^(k)+y^(-k)-2)-(1-(-1)^k)(y^(k)+y^(-k)-y-1/y))/(y-1/y)^2;

{aa+Nvec[[3]](bb-cc)>0,(cc-bb)Nvec[[3]]+2bb-aa>0,bb<cc,Cvec[[1]]>0,Cvec[[2]]<0,Cvec[[3]]<0}

Rank (1,1,2)

aa=3;bb=2;cc=3;Nvec={1,1,2};Cvec={1,-8/10,-1/10};
k=kk(aa+Nvec[[3]](bb-cc));
soH[{1,1,2}]={CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,1},y_]->-((2 (y^2))/(((-1+(y^2)))^2)),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,1,1},y_]->(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^kk) (1+(((-1))^kk)))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1},y_]->(1/8) kk y (((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2)))};
OmR[{1,1,2}]=OmS[{1,1,2}]+(( 
(1+(-1)^k)/2 (y^(-k+1)-y^(k-1)+1/4(k-1)(y+1/y)^2(y-1/y)+1/4(-1)^(k/2)(y-1/y)^3)
+(1-(-1)^k)/2(y^(k-1)-y^(1-k)-1/2(k-1)(y^2-1/y^2))))/(y-1/y)^3/(y+1/y);
{aa+Nvec[[3]](bb-cc)>0,(cc-bb)Nvec[[3]]+2bb-aa>0,bb<cc,Cvec[[1]]>0,Cvec[[2]]<0,Cvec[[3]]<0}

Rank (1,1,3)

aa=4;bb=2;cc=3;Nvec={1,1,3};Cvec={1,-7/10,-1/10};
k=kk(aa+Nvec[[3]](bb-cc));
soH[{1,1,3}]={CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,1},y_]->(1/2) y (-((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,1,1},y_]->(1/8) y (-(1/(((-1+y))^2))-(1/(((1+y))^2))+((2 (((-1))^kk))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{3,1,1},y_]->(1/18) y (-((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))+((2 (((-1))^kk) Cos[((kk π)/3)])/(1+(-1+y) y))-((2 (((-1))^kk) Cos[((2 kk π)/3)])/(1+y+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{2,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,1},{1,0,0}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,1},{1,0,0}},{2,1,1},y_]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1},y_]->((kk y (1+(y^2)))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,2,1,1},y_]->(1/16) y ((((((-1))^kk) kk)/(((-1+y))^2))-(kk/(((1+y))^2))-((2 (((-1))^kk) Sin[((kk π)/2)])/(1+(y^2)))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,1}},{1,1,1,1},y_]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,1},{1,0,0}},{1,1,1,1},y_]->0,CoulombH[{{0,0,1},{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1,1},y_]->(1/192) y (-((2 (((-1))^kk) (-1+(kk^2) (((-1+y))^2)-(-4+y) y))/(((-1+y))^4))+((2 (-1+(kk^2) (((1+y))^2)-y (4+y)))/(((1+y))^4)))};
OmR[{1,1,3}]=(1/576) (((96 (((-1))^kk) ((y^(-kk))+(y^kk)))/((((1/y)-y))^4))-((288 (((-y))^(4-kk)) (-1+(y^(2 kk))))/((((-1+(y^2)))^3) (1+(y^2))))+((192 (((-y))^(4-kk)) (1+(y^(2 kk))))/((((-1+(y^2)))^2) (1+(y^2)+(y^4))))+32 y (-((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))+(((E^((2 I kk π)/3))+(E^((4 I kk π)/3)))/(1-y+(y^2)))-(((E^((I kk π)/3))+(E^((5 I kk π)/3)))/(1+y+(y^2))))+18 y (((2 I (((-I))^(-kk)) (-1+(((-1))^kk)))/(1+(y^2)))+kk (((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))))+3 y (-((2 (((-1))^kk) (-1+(kk^2) (((-1+y))^2)+4 y-(y^2)))/(((-1+y))^4))+((2 (-1-4 y-(y^2)+(kk^2) (((1+y))^2)))/(((1+y))^4)))+576 OmS[{1,1,3}]);
(* expected answer, up to positive powers of y:*)
OmPol[{1,1,Nvec[[3]]}]=OmS[{1,1,Nvec[[3]]}]+(-1)^(k+Nvec[[3]]+1)y^(-k+Nvec[[3]]^2+1)/(1-y^2)^2/Product[1-y^(2i),{i,2,Nvec[[3]]}]; 
{aa+Nvec[[3]](bb-cc)>0,(cc-bb)Nvec[[3]]+2bb-aa>0,bb<cc,Cvec[[1]]>0,Cvec[[2]]<0,Cvec[[3]]<0,Sum[Nvec[[i]] Cvec[[i]],{i,3}]==0}

Rank (1,1,4)

aa=5;bb=2;cc=3;Nvec={1,1,4};Cvec={1,-6/10,-1/10};
k=kk(aa+Nvec[[3]](bb-cc));
soH[{1,1,4}]={CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,1,1},y]->(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^(7 kk)) (1+(((-1))^kk)))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{3,1,1},y]->(1/18) y (-(1/(((-1+y))^2))+(1/(((1+y))^2))+(((E^((20 I kk π)/3))+(E^((40 I kk π)/3)))/(1-y+(y^2)))-(((E^((10 I kk π)/3))+(E^((50 I kk π)/3)))/(1+y+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{4,1,1},y]->(1/64) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((I (((-1))^((3 (1+kk))/4)) (1+(I^kk)))/(((((-1))^(1/4))-y) (-1+(((-1))^(1/4)) y)))-((I (((-1))^((3-kk)/4)) (1+(I^kk)))/(((((-1))^(1/4))+y) (1+(((-1))^(1/4)) y)))-(((((-1))^((3+kk)/4)) (1+(I^(3 kk))))/(((((-1))^(3/4))-y) (-1+(((-1))^(3/4)) y)))+(((((-1))^(-(3/4) (-1+kk))) (1+(I^(3 kk))))/(((((-1))^(3/4))+y) (1+(((-1))^(3/4)) y)))-(((((-I))^(-kk)) (1+(((-1))^kk)))/(1+(y^2)))-(((I^kk) (1+(I^(2 kk))))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{3,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,2}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,2}},{2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,1},{1,0,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,1},{1,0,0}},{2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,1},{1,0,0}},{3,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1},y]->(3/8) kk y (((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,2,1,1},y]->((kk (y^2))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,3,1,1},y]->(1/72) y (((2 (((-1))^kk) kk)/(((-1+y))^2))+((2 kk)/(((1+y))^2))+((6 (((-1))^((1/3)+kk)) (E^(-(2/3) I kk π)) (-1+(E^((4 I kk π)/3))))/((-2+(((-1))^(1/3))) (((-1+(((-1))^(2/3))))^2) ((((-1))^(1/3))+y) (1+(((-1))^(1/3)) y)))+((6 (((-1))^((2/3)+kk)) (E^((I kk π)/3)) (-1+(E^(-(2/3) I kk π))))/((((1+(((-1))^(1/3))))^2) (2+(((-1))^(2/3))) ((((-1))^(2/3))+y) (1+(((-1))^(2/3)) y)))-((2 (((-1))^((1/3) (1+2 kk))) (-1+(((-1))^(2 kk/3))))/((-1+(((-1))^(2/3))) (1-y+(y^2))))-((2 (((-1))^((2+kk)/3)) (-1+(((-1))^(4 kk/3))))/((1+(((-1))^(1/3))) (1+y+(y^2))))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{2,2,1,1},y]->(1/256) y (((2 (((-1))^kk) kk)/(((-1+y))^2))+((2 kk)/(((1+y))^2))+(((((-1))^kk) (((-I))^(-kk)) (-2 I (-1+(I^(2 kk))) y+(1+(I^(2 kk))) kk (1+(y^2))))/(((1+(y^2)))^2))+(((I^kk) (2 I (-1+(I^(2 kk))) y+(1+(I^(2 kk))) kk (1+(y^2))))/(((1+(y^2)))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,1}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,1}},{1,2,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,2}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,1},{1,0,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,1},{1,0,0}},{1,2,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1,1},y]->-(((y^2) (2 (y^2)+(kk^2) (((-1+(y^2)))^2)))/(6 (((-1+(y^2)))^4))),CoulombH[{{0,0,1},{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,2,1,1},y]->(1/256) y (-((2 (((-I))^(-kk)) (1+(((-1))^kk)))/(1+(y^2)))-((2 (I^kk) (1+(I^(2 kk))))/(1+(y^2)))-((2 (((-1))^kk) ((kk^2) (((-1+y))^2)-2 (1-3 y+(y^2))))/(((-1+y))^4))-((2 ((kk^2) (((1+y))^2)-2 (1+3 y+(y^2))))/(((1+y))^4))),CoulombH[{{0,0,1},{0,0,1},{0,0,1},{0,1,0},{1,0,1}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,0,1},{0,1,1},{1,0,0}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1,1,1},y]->((kk y (((2 (((-1))^kk) (-4+(kk^2) (((-1+y))^2)+14 y-4 (y^2)))/(((-1+y))^4))+((2 ((kk^2) (((1+y))^2)-2 (2+7 y+2 (y^2))))/(((1+y))^4))))/4608)};
OmR[{1,1,4}]=(((((-1))^(5-kk)) (-(y^(-kk))+(y^kk)))/(24 (((-(1/y)+y))^5)))+(((((-1))^(3-kk)) (-(y^(-kk))+(y^kk)))/(8 (-(1/y)+y) (((-(1/(y^2))+(y^2)))^2)))+(((((-1))^(3-kk)) (-(y^(-kk))+(y^kk)))/(3 (((-(1/y)+y))^2) (-(1/(y^3))+(y^3))))+(((((-1))^(4-kk)) ((y^(-kk))+(y^kk)))/(4 (((-(1/y)+y))^3) (-(1/(y^2))+(y^2))))+(((((-1))^(2-kk)) ((y^(-kk))+(y^kk)))/(4 (-(1/y)+y) (-(1/(y^4))+(y^4))))+(1/64) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((I (((-1))^((3 (1+kk))/4)) (1+(I^kk)))/(((((-1))^(1/4))-y) (-1+(((-1))^(1/4)) y)))-((I (((-1))^((3-kk)/4)) (1+(I^kk)))/(((((-1))^(1/4))+y) (1+(((-1))^(1/4)) y)))-(((((-1))^((3+kk)/4)) (1+(I^(3 kk))))/(((((-1))^(3/4))-y) (-1+(((-1))^(3/4)) y)))+(((((-1))^(-(3/4) (-1+kk))) (1+(I^(3 kk))))/(((((-1))^(3/4))+y) (1+(((-1))^(3/4)) y)))-(((((-I))^(-kk)) (1+(((-1))^kk)))/(1+(y^2)))-(((I^kk) (1+(I^(2 kk))))/(1+(y^2))))+(1/72) y (((2 (((-1))^kk) kk)/(((-1+y))^2))+((2 kk)/(((1+y))^2))+((6 (((-1))^((1/3)+kk)) (E^(-(2/3) I kk π)) (-1+(E^((4 I kk π)/3))))/((-2+(((-1))^(1/3))) (((-1+(((-1))^(2/3))))^2) ((((-1))^(1/3))+y) (1+(((-1))^(1/3)) y)))+((6 (((-1))^((2/3)+kk)) (E^((I kk π)/3)) (-1+(E^(-(2/3) I kk π))))/((((1+(((-1))^(1/3))))^2) (2+(((-1))^(2/3))) ((((-1))^(2/3))+y) (1+(((-1))^(2/3)) y)))-((2 (((-1))^((1/3) (1+2 kk))) (-1+(((-1))^(2 kk/3))))/((-1+(((-1))^(2/3))) (1-y+(y^2))))-((2 (((-1))^((2+kk)/3)) (-1+(((-1))^(4 kk/3))))/((1+(((-1))^(1/3))) (1+y+(y^2)))))+(1/256) y (((2 (((-1))^kk) kk)/(((-1+y))^2))+((2 kk)/(((1+y))^2))+(((((-1))^kk) (((-I))^(-kk)) (-2 I (-1+(I^(2 kk))) y+(1+(I^(2 kk))) kk (1+(y^2))))/(((1+(y^2)))^2))+(((I^kk) (2 I (-1+(I^(2 kk))) y+(1+(I^(2 kk))) kk (1+(y^2))))/(((1+(y^2)))^2)))+(1/256) y (-((2 (((-I))^(-kk)) (1+(((-1))^kk)))/(1+(y^2)))-((2 (I^kk) (1+(I^(2 kk))))/(1+(y^2)))-((2 (((-1))^kk) ((kk^2) (((-1+y))^2)-2 (1-3 y+(y^2))))/(((-1+y))^4))-((2 ((kk^2) (((1+y))^2)-2 (1+3 y+(y^2))))/(((1+y))^4)))+((kk y ((((((-1))^kk) (-4+(kk^2) (((-1+y))^2)+14 y-4 (y^2)))/(((-1+y))^4))+(((kk^2) (((1+y))^2)-2 (2+7 y+2 (y^2)))/(((1+y))^4))))/2304)+OmS[{1,1,4}];
(* up to positive powers of y: *)
OmPol[{1,1,Nvec[[3]]}]=OmS[{1,1,Nvec[[3]]}]+(-1)^(k+Nvec[[3]]+1)y^(-k+Nvec[[3]]^2+1)/(1-y^2)^2/Product[1-y^(2i),{i,2,Nvec[[3]]}]; 
{aa+Nvec[[3]](bb-cc)>0,(cc-bb)Nvec[[3]]+2bb-aa>0,bb<cc,Cvec[[1]]>0,Cvec[[2]]<0,Cvec[[3]]<0,Sum[Nvec[[i]] Cvec[[i]],{i,n}]==0}

Rank (1,2,2)

aa=3;bb=2;cc=3;Nvec={1,2,2};Cvec={2,-29/10,19/10};
soH[{1,2,2}]={CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,1},y_]->-((2 (y^2))/(((-1+(y^2)))^2)),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,2,1},y_]->(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^(5 kk)) (1+(((-1))^kk)))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,1,1},y_]->(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^(7 kk)) (1+(((-1))^kk)))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,2,1},y_]->(1/32) y (-(1/(((-1+y))^2))+(1/(((1+y))^2))+(((I^(4 kk)) ((1+(I^(8 kk))) y-2 I (-1+(I^(8 kk))) kk (1+(y^2))))/(((1+(y^2)))^2))+(((1+(I^(8 kk))) y+2 I (-1+(I^(8 kk))) kk (1+(y^2)))/(((1+(y^2)))^2))),CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,2,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{1,1,0}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{1,1,0}},{2,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{1,1,1}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,1},{1,1,0}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,2,1},{1,0,0}},{1,1,1},y_]->0,CoulombH[{{0,1,0},{0,1,1},{1,0,1}},{1,1,1},y_]->0,CoulombH[{{0,1,0},{0,1,2},{1,0,0}},{1,1,1},y_]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1},y_]->(1/8) kk y (((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,2,1},y_]->((kk (y^2))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,1,0}},{1,1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{1,1,1,1},y_]->(1/8) kk y (((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))),CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{2,1,1,1},y_]->((kk (y^2))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,1}},{1,1,1,1},y_]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,1},{1,0,0}},{1,1,1,1},y_]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{1,1,1,1,1},y_]->-((y ((y^3)+2 (kk^2) y (((-1+(y^2)))^2)))/(2 (((-1+(y^2)))^4)))};
OmR[{1,2,2}]=OmS[{1,2,2}]+(y-1/y)^(-4)(y+1/y)^(-2)(-kk^2 y^4-kk^2/y^4+2kk^2-2kk+y^(2-4kk)+y^(4kk-2)+kk y^4+kk/y^4-y^2-1/y^2);

Rank (2,2,2)

aa=3;bb=2;cc=3;
Nvec={2,2,2};
Cvec={1,-29/10,19/10};
OmR[{2,2,2}]=((3 kk y (1+(y^4)))/(4 (((-1+(y^2)))^2) (1+(y^2))))+(((((-1))^(2-8 kk)) (-(1/y)+y) ((y^(-16 kk))+(y^(16 kk))))/(8 (((-(1/(y^2))+(y^2)))^3)))+(((((-1))^(4-8 kk)) ((y^(-16 kk))+(y^(16 kk))))/(8 (((-(1/y)+y))^3) (-(1/(y^2))+(y^2))))+(((((-1))^(4-8 kk)) (4+(y^(-16 kk))+(y^(16 kk))))/(8 (((-(1/y)+y))^3) (-(1/(y^2))+(y^2))))+((3 (((-1))^(3-8 kk)) (-(y^(-16 kk))+2 (y^(-4 kk))-2 (y^(4 kk))+(y^(16 kk))))/(8 (-(1/y)+y) (((-(1/(y^2))+(y^2)))^2)))+(((((-1))^(4-8 kk)) (-4+(y^(-16 kk))+4 (y^(-4 kk))+4 (y^(4 kk))+(y^(16 kk))))/(8 (((-(1/y)+y))^3) (-(1/(y^2))+(y^2))))+(((((-1))^(5-8 kk)) (-(y^(-16 kk))-4 (y^(-8 kk))+6 (y^(-4 kk))-6 (y^(4 kk))+4 (y^(8 kk))+(y^(16 kk))))/(8 (((-(1/y)+y))^5)))+((kk y (1+(y^2)) (7-3 (((-1))^kk)+2 (-67+3 (((-1))^kk)) (y^2)+(7-3 (((-1))^kk)) (y^4)+50 (kk^2) (((-1+(y^2)))^2)))/(48 (((-1+(y^2)))^4)))-((y (-1+(((-1))^kk)-2 (4+(((-1))^kk)) (y^4)+(-1+(((-1))^kk)) (y^8)+2 (kk^2) (((-1+(y^4)))^2)))/(8 (((-1+(y^2)))^4) (1+(y^2))))+(1/256) y (((4 (3+4 (((-1))^kk)))/(1+(y^2)))+(1/(((-1+y))^4))2 (-1+2 (((-1))^kk)-4 (I^(7 kk))-4 (I^(9 kk))-28 (kk^2) (((-1+y))^2)+(9-4 (((-1))^kk)+8 (I^(7 kk))+8 (I^(9 kk))) y+(-1+2 (((-1))^kk)-4 (I^(7 kk))-4 (I^(9 kk))) (y^2))+(1/(((1+y))^4))2 (-1+2 (((-1))^kk)-2 (I^(29 kk))-2 (I^(31 kk))-2 (I^(49 kk))-2 (I^(51 kk))+(-9+4 (((-1))^kk)-4 (I^(29 kk))-4 (I^(31 kk))-4 (I^(49 kk))-4 (I^(51 kk))) y+(-1+2 (((-1))^kk)-2 (I^(29 kk))-2 (I^(31 kk))-2 (I^(49 kk))-2 (I^(51 kk))) (y^2)-28 (kk^2) (((1+y))^2)))+(1/256) y (((4 (1+4 (((-1))^kk)))/(1+(y^2)))+(1/(((-1+y))^4))2 (1+2 (((-1))^kk)-4 (I^(27 kk))-4 (I^(29 kk))-28 (kk^2) (((-1+y))^2)+(3-4 (((-1))^kk)+8 (I^(27 kk))+8 (I^(29 kk))) y+(1+2 (((-1))^kk)-4 (I^(27 kk))-4 (I^(29 kk))) (y^2))+(1/(((1+y))^4))2 (1+2 (((-1))^kk)-2 (I^(49 kk))-2 (I^(51 kk))-2 (I^(69 kk))-2 (I^(71 kk))+(-3+4 (((-1))^kk)-4 (I^(49 kk))-4 (I^(51 kk))-4 (I^(69 kk))-4 (I^(71 kk))) y+(1+2 (((-1))^kk)-2 (I^(49 kk))-2 (I^(51 kk))-2 (I^(69 kk))-2 (I^(71 kk))) (y^2)-28 (kk^2) (((1+y))^2)))+(1/256) y ((6/(((-1+y))^2))+(6/(((1+y))^2))-((6 (1+6 (y^2)+(y^4)))/(((1+(y^2)))^3))+((-32 I (-1+(I^(32 kk))) kk y (1+(y^2))-128 (-1+(I^(32 kk))) (kk^2) (((1+(y^2)))^2)+(-7+(I^(32 kk))) (1+6 (y^2)+(y^4)))/(((1+(y^2)))^3)))+((kk y (1+(y^2)) OmS[{1,1,1}])/(((-1+(y^2)))^2))+(((((-1))^(3-2 kk)) (-(y^(-6 kk))+(y^(-4 kk))-(y^(4 kk))+(y^(6 kk))) (-((2 (y^2))/(((-1+(y^2)))^2))+OmS[{1,1,1}]))/(((-(1/y)+y))^3))+(((-(1/y)+y) (OmS[{1,1,1}]))/(2 (-(1/(y^2))+(y^2))))-(1/(2 (-(1/(y^2))+(y^2))))(-(1/y)+y) ((((((-1))^(2-2 kk)) (((((y^2)))^(-4 kk))+((((y^2)))^(4 kk))))/(((-(1/(y^2))+(y^2)))^2))+OmS[{1,1,1}])+(1/4) y (-((2 (((-1))^kk))/(((-1+y))^2))+(2/(((1+y))^2))) OmS[{1,1,2}]+(1/(((-(1/y)+y))^2))(((-1))^(2-5 kk)) ((y^(-5 kk))+(y^(5 kk))) (((kk y (1+(((-1))^kk)+2 (-1+(((-1))^kk)) y+(1+(((-1))^kk)) (y^2)))/(8 (((-1+(y^2)))^2)))+(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^kk) (1+(((-1))^kk)))/(1+(y^2))))+OmS[{1,1,2}])+(1/2) y (-((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))) OmS[{1,2,1}]+(1/(((-(1/y)+y))^2))(((-1))^(2+5 kk)) ((y^(-5 kk))+(y^(5 kk))) (((kk y (1+(((-1))^kk)+2 (-1+(((-1))^kk)) y+(1+(((-1))^kk)) (y^2)))/(8 (((-1+(y^2)))^2)))+(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^kk) (1+(((-1))^kk)))/(1+(y^2))))+OmS[{1,2,1}])-((2 (y^2) OmS[{2,1,1}])/(((-1+(y^2)))^2))+(((((-1))^(2-2 kk)) ((y^(-6 kk))+(y^(6 kk))) (((kk y (1+(y^2)))/(2 (((-1+(y^2)))^2)))+(1/16) y (-(2/(((-1+y))^2))-(2/(((1+y))^2))+((4 (((-1))^kk))/(1+(y^2))))+OmS[{2,1,1}]))/(((-(1/y)+y))^2))+(((((-1))^(1-2 kk)) (-(y^(-2 kk))+(y^(2 kk))) OmS[{2,1,2}])/(-(1/y)+y))+(((((-1))^(1+2 kk)) (-(y^(-2 kk))+(y^(2 kk))) OmS[{2,2,1}])/(-(1/y)+y))+OmS[{2,2,2}];
soH[{2,2,2}]={CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,1},y]->-((2 (y^2))/(((-1+(y^2)))^2)),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,1,2},y]->(1/16) y (-(2/(((-1+y))^2))-(2/(((1+y))^2))+((4 (((-1))^kk))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,2,1},y]->(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^kk) (1+(((-1))^kk)))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{1,2,2},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,1,1},y]->(1/16) y (-((2 (((-1))^kk))/(((-1+y))^2))-(2/(((1+y))^2))+((2 (I^kk) (1+(((-1))^kk)))/(1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,1,2},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,2,1},y]->(1/64) y (-((8 y)/(((-1+(y^2)))^2))+((4 (1+(I^(8 kk))) y)/(((1+(y^2)))^2))),CoulombH[{{0,0,1},{0,1,0},{1,0,0}},{2,2,2},y]->(1/256) y ((6/(((-1+y))^2))+(6/(((1+y))^2))-((6 (1+6 (y^2)+(y^4)))/(((1+(y^2)))^3))+((-32 I (-1+(I^(32 kk))) kk y (1+(y^2))-128 (-1+(I^(32 kk))) (kk^2) (((1+(y^2)))^2)+(-7+(I^(32 kk))) (1+6 (y^2)+(y^4)))/(((1+(y^2)))^3))),CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,1}},{1,2,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,1,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,1,0}},{2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,1,1}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{2,0,1}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{2,0,1}},{1,2,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{2,1,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{2,1,0}},{2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{2,1,1}},{1,1,1},y]->-((2 (y^2))/(((-1+(y^2)))^2)),CoulombH[{{0,0,1},{0,1,1},{1,1,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,1},{2,1,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,2,1},{1,0,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,2,1},{1,0,0}},{1,1,2},y]->0,CoulombH[{{0,0,1},{1,0,0},{1,2,0}},{1,1,1},y]->0,CoulombH[{{0,0,1},{1,0,0},{1,2,0}},{2,1,1},y]->0,CoulombH[{{0,0,1},{1,0,0},{1,2,1}},{1,1,1},y]->(1/2) y (-((((-1))^kk)/(((-1+y))^2))+(1/(((1+y))^2))),CoulombH[{{0,0,1},{1,0,1},{1,2,0}},{1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,1},{1,0,1}},{1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,1},{2,0,1}},{1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,2},{1,0,0}},{1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,2},{1,0,0}},{1,1,2},y]->0,CoulombH[{{0,1,0},{1,0,0},{1,0,2}},{1,1,1},y]->0,CoulombH[{{0,1,0},{1,0,0},{1,0,2}},{2,1,1},y]->0,CoulombH[{{0,1,0},{1,0,0},{1,1,2}},{1,1,1},y]->(1/4) y (-((2 (((-1))^kk))/(((-1+y))^2))+(2/(((1+y))^2))),CoulombH[{{0,1,0},{1,0,2},{1,1,0}},{1,1,1},y]->0,CoulombH[{{0,1,1},{1,0,1},{1,1,0}},{1,1,1},y]->0,CoulombH[{{0,1,2},{1,0,0},{1,1,0}},{1,1,1},y]->0,CoulombH[{{0,2,1},{1,0,0},{1,0,1}},{1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,1},y]->((kk y (1+(((-1))^kk)+2 (-1+(((-1))^kk)) y+(1+(((-1))^kk)) (y^2)))/(8 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,1,2},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,2,1},y]->((kk (y^2))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0}},{1,1,2,2},y]->((kk y (1+(y^4)))/(4 (((-1+(y^2)))^2) (1+(y^2)))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,1,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{2,1,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{1,0,0},{1,2,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{1,1,1,1},y]->((kk y (1+(((-1))^kk)+2 (-1+(((-1))^kk)) y+(1+(((-1))^kk)) (y^2)))/(8 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{1,1,1,2},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{2,1,1,1},y]->((kk (y^2))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{2,1,1,2},y]->((kk y (1+(y^4)))/(4 (((-1+(y^2)))^2) (1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,1}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,0},{2,0,1}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,1},{1,0,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,1},{1,0,0}},{1,1,1,2},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,0,0}},{1,1,1,1},y]->((kk y (1+(y^2)))/(2 (((-1+(y^2)))^2))),CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,0,0}},{1,2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,0,0}},{2,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,0,0}},{2,2,1,1},y]->((kk y (1+(y^4)))/(4 (((-1+(y^2)))^2) (1+(y^2)))),CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,0,1}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,0,1}},{1,2,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,1,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,1,0}},{2,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{1,0,0},{1,1,1}},{1,1,1,1},y]->((kk y (1+(y^2)))/(((-1+(y^2)))^2)),CoulombH[{{0,0,1},{0,1,0},{1,0,1},{1,1,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,1},{1,0,0},{1,1,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,2,1},{1,0,0},{1,0,0}},{1,1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,0},{1,0,0},{1,0,2}},{1,1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,1},{1,0,0},{1,0,1}},{1,1,1,1},y]->0,CoulombH[{{0,1,0},{0,1,2},{1,0,0},{1,0,0}},{1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{1,1,1,1,1},y]->-(((y^2) ((y^2)+2 (kk^2) (((-1+(y^2)))^2)))/(2 (((-1+(y^2)))^4))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{0,1,0},{1,0,0}},{1,1,1,1,2},y]->-((y (-1+(((-1))^kk)-2 (4+(((-1))^kk)) (y^4)+(-1+(((-1))^kk)) (y^8)+2 (kk^2) (((-1+(y^4)))^2)))/(8 (((-1+(y^2)))^4) (1+(y^2)))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0},{1,0,0}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0},{1,0,0}},{1,1,2,1,1},y]->(1/256) y (((4 (1+4 (((-1))^kk)))/(1+(y^2)))+(1/(((-1+y))^4))2 (1+2 (((-1))^kk)-4 (I^(27 kk))-4 (I^(29 kk))-28 (kk^2) (((-1+y))^2)+(3-4 (((-1))^kk)+8 (I^(27 kk))+8 (I^(29 kk))) y+(1+2 (((-1))^kk)-4 (I^(27 kk))-4 (I^(29 kk))) (y^2))+(1/(((1+y))^4))2 (1+2 (((-1))^kk)-2 (I^(49 kk))-2 (I^(51 kk))-2 (I^(69 kk))-2 (I^(71 kk))+(-3+4 (((-1))^kk)-4 (I^(49 kk))-4 (I^(51 kk))-4 (I^(69 kk))-4 (I^(71 kk))) y+(1+2 (((-1))^kk)-2 (I^(49 kk))-2 (I^(51 kk))-2 (I^(69 kk))-2 (I^(71 kk))) (y^2)-28 (kk^2) (((1+y))^2))),CoulombH[{{0,0,1},{0,0,1},{0,1,0},{1,0,0},{1,1,0}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0},{1,0,0}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0},{1,0,0}},{2,1,1,1,1},y]->(1/256) y (((4 (3+4 (((-1))^kk)))/(1+(y^2)))+(1/(((-1+y))^4))2 (-1+2 (((-1))^kk)-4 (I^(7 kk))-4 (I^(9 kk))-28 (kk^2) (((-1+y))^2)+(9-4 (((-1))^kk)+8 (I^(7 kk))+8 (I^(9 kk))) y+(-1+2 (((-1))^kk)-4 (I^(7 kk))-4 (I^(9 kk))) (y^2))+(1/(((1+y))^4))2 (-1+2 (((-1))^kk)-2 (I^(29 kk))-2 (I^(31 kk))-2 (I^(49 kk))-2 (I^(51 kk))+(-9+4 (((-1))^kk)-4 (I^(29 kk))-4 (I^(31 kk))-4 (I^(49 kk))-4 (I^(51 kk))) y+(-1+2 (((-1))^kk)-2 (I^(29 kk))-2 (I^(31 kk))-2 (I^(49 kk))-2 (I^(51 kk))) (y^2)-28 (kk^2) (((1+y))^2))),CoulombH[{{0,0,1},{0,1,0},{0,1,0},{1,0,0},{1,0,1}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,1,0},{0,1,1},{1,0,0},{1,0,0}},{1,1,1,1,1},y]->0,CoulombH[{{0,0,1},{0,0,1},{0,1,0},{0,1,0},{1,0,0},{1,0,0}},{1,1,1,1,1,1},y]->((kk y (1+(y^2)) (7-3 (((-1))^kk)+2 (-67+3 (((-1))^kk)) (y^2)+(7-3 (((-1))^kk)) (y^4)+50 (kk^2) (((-1+(y^2)))^2)))/(48 (((-1+(y^2)))^4)))};

Evaluate the Poincare polynomial

Mat=(({{0, aa, -cc}, {-aa, 0, bb}, {cc, -bb, 0}}));
$QuiverMultiplier=kk;
$QuiverMultiplierAssumption={kk∈Integers,kk>0};
$QuiverNoLoop=False;
$QuiverTestLoop=True;
$QuiverDisplayCoulombH=True;

{QPoinca,CoulombHRule}=CoulombBranchFormula[Mat,Cvec,Nvec]

CoulombHRuleAny=Simplify[CoulombHRule,kk∈Integers]

Simplify[Series[Table[DropFugacity[QPoinca],{kk,1,8}],{y,0,50}]]

Simplify[Table[Series[(DropFugacity[QPoinca]/.OmS[gam_,t_]:>OmS[gam])-OmR[Nvec],{y,0,30}],{kk,1,8}]]