SharedHypersimplexStapledon.sagewsOpen in CoCalc
Author: Victor Reiner
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Description: Calculating Stapledon's h*-representations of S_n for hypersimplices
s = SymmetricFunctions(QQ).schur(); h = SymmetricFunctions(QQ).homogeneous(); e = SymmetricFunctions(QQ).elementary(); m = SymmetricFunctions(QQ).monomial();
s(m[2])
-s[1, 1] + s[2]
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((s[3]+s[1,1,1]).plethysm(h[2]));
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