Mathematics 501
A. Hulpke
Homework (due Nov 8)
48) Let f (x) = Q[x] be a monic (i.e the coefficient of the largest power of x occurring is 1) polynomial. We
define the discriminant of f as
D( f ) = ∆2 ( f ) = ∏(αi − α j )
i6= j
where the {αi } are the roots of f , including multiplicities.
a) Show that D( f ) = 0 if and only f has multiple roots.
b) Show that if f (x) ∈ Z[x] then D( f ) ∈ Z.
c) Show that if f (x) ∈ Z[x] has no multiple roots that there are only finitely many primes p such that f has
multiple roots modulo p.
49) Write the polynomial
2x31 x32 x3
+ 2x31 x32 x4 + 2x31 x2 x33 + 2x31 x2 x34 + 2x31 x33 x4 + 2x31 x3 x34 + 2x1 x32 x33 + 2x1 x32 x34 + 2x1 x33 x34
+ 2x32 x33 x4 + 2x32 x3 x34 + 2x2 x33 x34 + 2x21 x2 x3 + 2x21 x2 x4 + 2x21 x3 x4 + 2x1 x22 x3 + 2x1 x22 x4
+ 2x1 x2 x23 + 2x1 x2 x24 + 2x1 x23 x4 + 2x1 x3 x24 + 2x22 x3 x4 + 2x2 x23 x4 + 2x2 x3 x24
as a polynomial in the elementary symmetric polynomials.
You might want to use a computer to help with tedious arithmetic. Here are GAP commands to create the
polynomial and the elementary symmetric polynomials:
gap> r:=PolynomialRing(Rationals,4);
gap> s:=SymmetricGroup(4);
gap> v:=IndeterminatesOfPolynomialRing(r);
[ x_1, x_2, x_3, x_4 ]
gap> z:=v[1]*v[2]^3*v[3]^3; y:=v[1]*v[2]*v[3]^2;
gap> p:=Sum(Orbit(s,z+y,OnIndeterminates));
2*x_1^3*x_2^3*x_3+2*x_1^3*x_2^3*x_4+2*x_1^3*x_2*x_3^3+[...]
gap> e1:=Sum(Orbit(s,v[1],OnIndeterminates));
x_1+x_2+x_3+x_4
gap> e2:=Sum(Orbit(s,v[1]*v[2],OnIndeterminates));;
x_1*x_2+x_1*x_3+x_1*x_4+x_2*x_3+x_2*x_4+x_3*x_4
gap> e3:=Sum(Orbit(s,v[1]*v[2]*v[3],OnIndeterminates));;
gap> e4:=Sum(Orbit(s,v[1]*v[2]*v[3]*v[4],OnIndeterminates));;
50∗ ) Let λ = (λ1 , λ2 , . . .) |= n. We consider sequences (called lattice permutations or Yamanouchi words or
ballot sequences) of the form a1 a2 · · · an in which ai occurs λi times and such that in any prefix a1 a2 · · · a j the
number of i’s occuring is at least as big as the number of i + 1’s.
For example for λ = (3, 2) there are five such sequences:
11122
11212
11221
12112
12121
Show that there are f λ such sequences.
51) The casual users of a library have placed books back on the shelves in the wrong position. We call the
minimum number of such misplacements the “disorder” of the library. Thus the disorder is the minimum
number of removal/insertion operations that will put the library back in order again. If we consider the
misarranged library to be given by a permutation π , how can one describe the disorder by a property of π
(or the tableaux obtained from π via the RSK algorithm)?
52) Calculate the Schur polynomial for the partition (3, 1).