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Calculating Stapledon's h*-representations of S_n for hypersimplices

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s = SymmetricFunctions(QQ).schur(); h = SymmetricFunctions(QQ).homogeneous(); e = SymmetricFunctions(QQ).elementary(); m = SymmetricFunctions(QQ).monomial();
A0=h[4]; A1=h[4]+h[3,1]+h[2,2]; A2=A1; A3=A0;
notbad3by3minor=(A2.itensor(A2)).itensor(A3)-(A1.itensor(A3)).itensor(A3); s(notbad3by3minor)
s[1, 1, 1, 1] + 8*s[2, 1, 1] + 10*s[2, 2] + 18*s[3, 1] + 11*s[4]
bad3by3minor=(A1.itensor(A1)).itensor(A2)-A1.itensor(A3)-A2.itensor(A2); s(bad3by3minor)
29*s[1, 1, 1, 1] + 124*s[2, 1, 1] + 103*s[2, 2] + 172*s[3, 1] + 76*s[4]
h(s[3,2,2])
h[3, 2, 2] - h[3, 3, 1] - h[4, 2, 1] + h[4, 3] + h[5, 1, 1] - h[5, 2]
load("burnside-solver.sage")
Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python3.10/site-packages/smc_sagews/sage_server.py", line 1248, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> File "/cocalc/lib/python3.10/site-packages/smc_sagews/sage_salvus.py", line 3923, in load '__kwds': kwds File "<string>", line 1, in <module> File "sage/misc/persist.pyx", line 145, in sage.misc.persist.load (build/cythonized/sage/misc/persist.c:2483) sage.repl.load.load(filename, globals()) File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/repl/load.py", line 272, in load exec(preparse_file(f.read()) + "\n", globals) File "<string>", line 68 if i:= p.get_values(v[G]): ^ SyntaxError: invalid syntax
Ashriek1=(h[3]+h[2,1]);Ashriek1 Ashriek2=Ashriek1.itensor(Ashriek1)-h[3];Ashriek2 Ashriek3=Ashriek1.itensor(Ashriek2)-Ashriek1;Ashriek3 Ashriek4=Ashriek1.itensor(Ashriek3)-Ashriek2;Ashriek4
h[2, 1] + h[3] h[1, 1, 1] + 3*h[2, 1] 7*h[1, 1, 1] + 5*h[2, 1] - h[3] 32*h[1, 1, 1] + 6*h[2, 1] - h[3]
s[3, 2] + s[4, 1] + s[5] s[2, 1, 1, 1] + 2*s[2, 2, 1] + 2*s[3, 1, 1] + 4*s[3, 2] + 4*s[4, 1] + 3*s[5] 2*s[1, 1, 1, 1, 1] + 8*s[2, 1, 1, 1] + 13*s[2, 2, 1] + 15*s[3, 1, 1] + 18*s[3, 2] + 16*s[4, 1] + 7*s[5]
phi1=OSshriek1;phi1;h(phi1) phi2=OSshriek2-s[2].inner_plethysm(OSshriek1);phi2;h(phi2) phi3=OSshriek3-s[3].inner_plethysm(OSshriek1)-phi1.itensor(phi2);phi3;h(phi3)
s[3, 2] + s[4, 1] + s[5] h[3, 2] s[2, 1, 1, 1] + s[3, 1, 1] h[2, 1, 1, 1] - 2*h[2, 2, 1] + h[3, 2] 2*s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + s[4, 1] 2*h[2, 2, 1] - h[3, 1, 1] - h[3, 2]
OSshriek1
s[3, 2] + s[4, 1] + s[5]
s[2].inner_plethysm(OSshriek1)
2*s[2, 2, 1] + s[3, 1, 1] + 4*s[3, 2] + 4*s[4, 1] + 3*s[5]
s[1,1,1]*s[2]
s[2, 1, 1, 1] + s[3, 1, 1]
lie2=s[1,1]; lie3=s[2,1]; lie4=s[3,1]+s[2,1,1]; lie5=s[4,1]+s[3,2]+s[3,1,1]+s[2,2,1]+s[2,1,1,1];
s(e[4].plethysm(lie2));
s[3, 2, 2, 1] + s[4, 1, 1, 1, 1]
s(h[4].plethysm(h[2]))
s[2, 2, 2, 2] + s[4, 2, 2] + s[4, 4] + s[6, 2] + s[8]
h(h[2].plethysm(h[2])); h(h[2].plethysm(h[3])); h(h[2].plethysm(h[4])); h(h[2].plethysm(h[5])); h(h[2].plethysm(h[6]));
h[2, 2] - h[3, 1] + h[4] h[4, 2] - h[5, 1] + h[6] h[4, 4] - h[5, 3] + h[6, 2] - h[7, 1] + h[8] h[6, 4] - h[7, 3] + h[8, 2] - h[9, 1] + h[10] h[6, 6] - h[7, 5] + h[8, 4] - h[9, 3] + h[10, 2] - h[11, 1] + h[12]
h[2, 2, 2] - 2*h[3, 2, 1] + 2*h[3, 3] + 2*h[4, 1, 1] - 2*h[4, 2] - 2*h[5, 1] + 2*h[6]
h((s[3]+s[1,1,1]).plethysm(h[3]));
h[3, 3, 3] - 2*h[4, 3, 2] + 2*h[4, 4, 1] + 2*h[5, 2, 2] - 2*h[5, 3, 1] - 2*h[5, 4] - 2*h[6, 2, 1] + 4*h[6, 3] + 2*h[7, 1, 1] - 2*h[7, 2] - 2*h[8, 1] + 2*h[9]
h((s[3]+s[1,1,1]).plethysm(h[4]));
h[4, 4, 4] - 2*h[5, 4, 3] + 2*h[5, 5, 2] + 2*h[6, 3, 3] - 2*h[6, 4, 2] - 2*h[6, 5, 1] + 2*h[6, 6] - 2*h[7, 3, 2] + 4*h[7, 4, 1] - 2*h[7, 5] + 2*h[8, 2, 2] - 2*h[8, 3, 1] - 2*h[8, 4] - 2*h[9, 2, 1] + 4*h[9, 3] + 2*h[10, 1, 1] - 2*h[10, 2] - 2*h[11, 1] + 2*h[12]
h((s[3]+s[1,1,1]).plethysm(h[5]));
h[5, 5, 5] - 2*h[6, 5, 4] + 2*h[6, 6, 3] + 2*h[7, 4, 4] - 2*h[7, 5, 3] - 2*h[7, 6, 2] + 2*h[7, 7, 1] - 2*h[8, 4, 3] + 4*h[8, 5, 2] - 2*h[8, 6, 1] - 2*h[8, 7] + 2*h[9, 3, 3] - 2*h[9, 4, 2] - 2*h[9, 5, 1] + 4*h[9, 6] - 2*h[10, 3, 2] + 4*h[10, 4, 1] - 2*h[10, 5] + 2*h[11, 2, 2] - 2*h[11, 3, 1] - 2*h[11, 4] - 2*h[12, 2, 1] + 4*h[12, 3] + 2*h[13, 1, 1] - 2*h[13, 2] - 2*h[14, 1] + 2*h[15]
def mthHypersimplexRep(n,k,m): rep=0; for subset in Subsets(range(1,n)): l=[0]; l.extend(sorted(subset)); l.extend([n]); tot=sum([i for i in subset]); card=subset.cardinality(); f(t,q)=1/(prod([(1-t*q^i) for i in subset])*(1-t*q^n)); qsum=taylor(f,(t,0),m-card)(1,q); coef=ZZ(qsum.coefficient(q,k*m-tot)); hrep=prod(s[l[i]-l[i-1]] for i in range(1,len(l))); rep=rep+coef*hrep; return(rep)
def wedgerep(n,i): rep=0; if i==0: rep=s[n]; elif i > n: rep=0 else: if i==n: rep=s[list(1 for j in range(n))] else: rep=s[[n-i]+list(1 for j in range(i))] + s[[n-i+1]+list(1 for j in range(i-1))] return(rep)
def StapledonHypHstar(n,k,i): rep=0; for m in range(i+1): rep=rep + (-1)^m * mthHypersimplexRep(n,k,i-m).itensor(wedgerep(n,m)); return(rep)
def StapledonVolRep(n,k): return(sum( StapledonHypHstar(n,k,i) for i in range(n)))
v63=h(StapledonVolRep(6,3)); v63;s(v63)
h[2, 2, 2] - 2*h[3, 2, 1] + 3*h[3, 3] + 2*h[4, 1, 1] - h[4, 2] - 2*h[5, 1] + 3*h[6] s[2, 2, 2] + 2*s[3, 3] + s[4, 1, 1] + 3*s[4, 2] + 2*s[5, 1] + 4*s[6]
v73=h(StapledonVolRep(7,3)); v73;
h[3, 2, 2] + 2*h[4, 3] + h[5, 2] + h[7]
v83=h(StapledonVolRep(8,3)); v83;
h[3, 3, 2] + h[4, 2, 2] + h[4, 4] + 2*h[5, 3] + h[6, 2] + h[8]
v93=h(StapledonVolRep(9,3)); v93;
h[3, 3, 3] + 2*h[4, 4, 1] + 3*h[5, 2, 2] - 2*h[5, 3, 1] - 2*h[6, 2, 1] + 6*h[6, 3] + 2*h[7, 1, 1] - h[7, 2] - 2*h[8, 1] + 3*h[9]
s(v93)-s(h[9]+h[7,2]+2*h[6,3]+2*h[5,4]+h[5,2,2]+2*h[4,3,2]+(h[3]+s[1,1,1]).plethysm(h[3]));
0
StapledonHypHstar(4,2,0); StapledonHypHstar(4,2,1); StapledonHypHstar(4,2,2); StapledonHypHstar(4,2,3);
h(StapledonHypHstar(5,2,0)); h(StapledonHypHstar(5,2,1)); h(StapledonHypHstar(5,2,2)); h(StapledonHypHstar(5,2,3));
h(StapledonHypHstar(6,2,0)); h(StapledonHypHstar(6,2,1)); h(StapledonHypHstar(6,2,2)); h(StapledonHypHstar(6,2,3)); h(StapledonHypHstar(6,2,4));
h(StapledonHypHstar(7,2,0)); h(StapledonHypHstar(7,2,1)); h(StapledonHypHstar(7,2,2)); h(StapledonHypHstar(7,2,3)); h(StapledonHypHstar(7,2,4)); h(StapledonHypHstar(8,2,0)); h(StapledonHypHstar(8,2,1)); h(StapledonHypHstar(8,2,2)); h(StapledonHypHstar(8,2,3)); h(StapledonHypHstar(8,2,4)); h(StapledonHypHstar(8,2,5)); h(StapledonHypHstar(9,2,0)); h(StapledonHypHstar(9,2,1)); h(StapledonHypHstar(9,2,2)); h(StapledonHypHstar(9,2,3)); h(StapledonHypHstar(9,2,4)); h(StapledonHypHstar(9,2,5));
h(StapledonHypHstar(10,2,0)); h(StapledonHypHstar(10,2,1)); h(StapledonHypHstar(10,2,2)); h(StapledonHypHstar(10,2,3)); h(StapledonHypHstar(10,2,4)); h(StapledonHypHstar(10,2,5));
h[10] h[8, 2] - h[9, 1] h[6, 4] h[6, 4] h[8, 2] h[10]
h(StapledonHypHstar(11,2,0)); h(StapledonHypHstar(11,2,1)); h(StapledonHypHstar(11,2,2)); h(StapledonHypHstar(11,2,3)); h(StapledonHypHstar(11,2,4)); h(StapledonHypHstar(11,2,5)); h(StapledonHypHstar(11,2,6));
h[11] h[9, 2] - h[10, 1] h[7, 4] h[6, 5] h[8, 3] h[10, 1] 0
h(StapledonHypHstar(6,3,0)); h(StapledonHypHstar(6,3,1)); h(StapledonHypHstar(6,3,2)); h(StapledonHypHstar(6,3,3)); h(StapledonHypHstar(6,3,4)); h(StapledonHypHstar(6,3,5));
h[6] h[3, 3] - h[5, 1] h[2, 2, 2] - 2*h[3, 2, 1] + h[3, 3] + 2*h[4, 1, 1] - h[4, 2] + h[6] h[3, 3] - h[5, 1] h[6] 0
h(StapledonVolRep(7,3)); h(StapledonHypHstar(7,3,0)); h(StapledonHypHstar(7,3,1)); h(StapledonHypHstar(7,3,2)); h(StapledonHypHstar(7,3,3)); h(StapledonHypHstar(7,3,4)); h(StapledonHypHstar(7,3,5)); h(StapledonHypHstar(7,3,6));
h[3, 2, 2] + 2*h[4, 3] + h[5, 2] + h[7] h[7] h[4, 3] - h[6, 1] h[3, 2, 2] - h[3, 3, 1] + h[4, 3] + h[5, 1, 1] - h[5, 2] + h[6, 1] h[3, 3, 1] - h[5, 1, 1] + h[5, 2] h[5, 2] 0 0
h(StapledonHypHstar(8,3,0)); h(StapledonHypHstar(8,3,1)); h(StapledonHypHstar(8,3,2)); h(StapledonHypHstar(8,3,3)); h(StapledonHypHstar(8,3,4)); h(StapledonHypHstar(8,3,5)); h(StapledonHypHstar(8,3,6));
h[8] h[5, 3] - h[7, 1] h[4, 2, 2] - h[5, 2, 1] + h[5, 3] + h[6, 1, 1] h[3, 3, 2] + h[7, 1] h[4, 4] + h[5, 2, 1] - h[6, 1, 1] + h[6, 2] - h[7, 1] h[7, 1] 0
StapledonHypHstar(9,3,0); StapledonHypHstar(9,3,1); StapledonHypHstar(9,3,2); StapledonHypHstar(9,3,3); StapledonHypHstar(9,3,4); StapledonHypHstar(9,3,5); StapledonHypHstar(9,3,6); StapledonHypHstar(9,3,7); StapledonHypHstar(9,3,8);
s[9] s[6, 3] + s[7, 2] s[4, 4, 1] + s[5, 2, 2] + s[5, 3, 1] + 2*s[5, 4] + s[6, 2, 1] + 3*s[6, 3] + s[7, 1, 1] + 3*s[7, 2] + 3*s[8, 1] + 2*s[9] s[3, 3, 3] + 2*s[4, 3, 2] + s[4, 4, 1] + 2*s[5, 2, 2] + 3*s[5, 3, 1] + 2*s[5, 4] + 3*s[6, 2, 1] + 4*s[6, 3] + 2*s[7, 1, 1] + 5*s[7, 2] + 4*s[8, 1] + 3*s[9] s[4, 4, 1] + s[5, 2, 2] + 2*s[5, 3, 1] + 3*s[5, 4] + 2*s[6, 2, 1] + 5*s[6, 3] + 4*s[7, 2] + 2*s[8, 1] + 2*s[9] s[6, 3] + s[7, 1, 1] + 2*s[7, 2] + 2*s[8, 1] + s[9] s[9] 0 0
StapledonHypHstar(10,3,0); StapledonHypHstar(10,3,1); StapledonHypHstar(10,3,2); StapledonHypHstar(10,3,3); StapledonHypHstar(10,3,4); StapledonHypHstar(10,3,5); StapledonHypHstar(10,3,6); StapledonHypHstar(10,3,7); StapledonHypHstar(10,3,8);
s[10] s[7, 3] + s[8, 2] s[5, 4, 1] + s[5, 5] + s[6, 2, 2] + s[6, 3, 1] + 3*s[6, 4] + s[7, 2, 1] + 3*s[7, 3] + s[8, 1, 1] + 3*s[8, 2] + 3*s[9, 1] + 2*s[10] s[4, 3, 3] + s[4, 4, 2] + 3*s[5, 3, 2] + 2*s[5, 4, 1] + s[5, 5] + 2*s[6, 2, 2] + 4*s[6, 3, 1] + 3*s[6, 4] + 4*s[7, 2, 1] + 6*s[7, 3] + 2*s[8, 1, 1] + 6*s[8, 2] + 5*s[9, 1] + 3*s[10] s[4, 4, 2] + 2*s[5, 3, 2] + 3*s[5, 4, 1] + 2*s[5, 5] + 2*s[6, 2, 2] + 4*s[6, 3, 1] + 6*s[6, 4] + 3*s[7, 2, 1] + 6*s[7, 3] + s[8, 1, 1] + 6*s[8, 2] + 4*s[9, 1] + 3*s[10] s[5, 5] + s[6, 3, 1] + 2*s[6, 4] + 2*s[7, 2, 1] + 4*s[7, 3] + s[8, 1, 1] + 4*s[8, 2] + 2*s[9, 1] + s[10] s[8, 2] + 2*s[9, 1] + 2*s[10] 0 0