function f (x1, . . . , xn) which is invariant under all permutations of the variables. We write Symn = Z[x1, . . . , xn]Sn for the ring of symmetric polynomials with integer coeﬃcients. Here Sn is the symmetric group, consisting of all permutations of the variables x1, . . . , xn. Examples er = i1<i2<···<ir xi1 xi2 · · · xir = rth elementary symmetric function. hr = i1≤i2≤···≤ir xi1 xi2 · · · xir = rth homogeneous symmetric function. Theorem (Fundamental Theorem) Symn = Z[e1, . . . , en] = Z[h1, . . . , hn]. The {er } and {hr } are algebraically independent over Z.
. . , αn), where αj ∈ Z≥0, deﬁne eα = eα1 eα2 · · · eαn , hα = hα1 hα2 · · · hαn . The sets {eα}, {hα} clearly span Symn . But they are not linearly independent. For instance, h(1,2) = h(2,1) , etc. Deﬁnition A sequence λ = (λ1, . . . , λn) is a partition of r (written as λ r) if λ1 ≥ λ2 · · · ≥ λn ≥ 0 and λj = r. Its length (λ) is the number of non-zero parts. Notice that any sequence α = (α1, . . . , αn) of non-negative integers can be permuted to a partition λ. Thus: Theorem The sets {eλ | λ a partition, λi ≤ n ∀i} and {hλ | λ a partition, (λ) ≤ n} are Z-bases of Symn .
αn), deﬁne aα(x1, . . . , xn) = w∈Sn sgn(w)w(xα). This is skew-symmetric, i.e., w(aα) = sgn(w)aα. In particular, aα = 0 unless all parts of α are distinct. So may as well assume that α1 > α2 > · · · > αn ≥ 0 and thus write α = λ + δ where δ = (n − 1, n − 2, . . . , 1, 0) and λ is a partition. Now aδ = det(xn−j i ) = i<j (xi − xj ) is a Vandermonde determinant. It divides aα = aλ+δ, and we deﬁne the Schur function sλ to be the quotient sλ(x1, . . . , xn) = aλ+δ(x1, . . . , xn) aδ(x1, . . . , xn) ∈ Symn . The function sλ is symmetric. If λ is not a partition, then either sλ = 0 or sλ = ±sµ where µ is a partition. So: Theorem {sλ(x1, . . . , xn) | λ a partition, (λ) ≤ n} is a Z-basis of Symn .
1 + · · · + xr n = rth power sum. For a sequence α = (α1, . . . , αn) set pα = pα1 pα2 · · · pαn . Then with Q Symn := Q ⊗ Z Symr we have: Theorem Q Symr = Q[p1, . . . , pn] where the {pr } are algebraically independent. This is false over Z.
λn) be a partition with (λ) ≤ n. The Jacobi–Trudi identity says that sλ = det(hλi −i+j )1≤i,j≤n = det hλ1 hλi −1 · · · hλ2+1 hλ2 · · · . . . . . . ... . This expresses Schur functions in terms of complete symmetric functions. There is a “dual” formula expressing sλ in terms of elementary symmetric functions.
G is a group homomorphism ρ : G → Aut C (V ) into the group of C-linear automorphisms of a C-vector space V . Equivalently, it is an algebra homomorphism CG → End C (V ) from the group algebra CG into the algebra of C-linear endomorphisms of V . Equivalently, it is a CG-module V . (The module viewpoint was ﬁrst emphasized by Emmy Noether.) Deﬁnition (Let V be a representation) V is irreducible if (0) is the only proper submodule. V is completely reducible (semisimple) if it is a direct sum of irreducible submodules. The trace character χV of V is deﬁned by χV (g) = trace ρ(g) for g ∈ G. This is constant on conjugacy classes of G.
ﬁnite group then the ﬁnite-dimensional ordinary representations are semisimple and are completely determined by their trace characters. The main problem is to compute the character table of values of the irreps on the conjugacy classes. Example. If G = Sn the conjugacy classes are indexed by partitions of n (cycle type theorem) and the character table is the matrix (χλ µ ) with rows and columns indexed by partitions, where pµ = λ χλ µ sλ. In other words, the transition matrix M(p, s) expressing the power sum basis in terms of the Schur function basis coincides with the character table of Sn. Remark. The modular irreps of Sn (over a ﬁeld of prime characteristic) are still in general unknown. We do not even know their dimensions!
(inﬁnite) Lie group G = GLn(C) of all invertible n × n complex matrices. Theorem 1. Finite dimensional polynomial representations of GLn are semisimple and are completely determined by their trace characters (equivalently, by their formal characters). 2. The irreducible representations are indexed by partitions λ with (λ) ≤ n. 3. The irreducible character χλ is given by the Schur function sλ(x1, . . . , xn). The fact that Schur functions compute the irreducible characters of both the symmetric and general linear groups is a manifestation of Schur–Weyl duality that links their rep theory.
an alternating sum of hµ’s. It is easy to ﬁnd a representation of GLn(C) whose character is hµ. Namely, let V = Cn with GLn(C) acting by matrix multiplication. Then the rth symmetric power Sr V is also a representation, and it is easy to see that hr = character of Sr V . It follows that if we deﬁne SµV = Sµ1 V ⊗ · · · ⊗ Sµn V then hµ = character of SµV . So the Jacobi–Trudi identity tells us that the ordinary representations of GLn(C) are determined by the symmetric powers of V .
that the ordinary representations of certain Lie groups govern the interactions of subatomic particles. This is nowadays formulated in terms of the standard model, which is based on Murray Gell-Mann’s “eightfold way”. The eightfold way stems from the adjoint (eight-dimensional) representation of the compact group SU3(C). This representation is isomorphic to the adjoint representation of GL3(C) in a certain sense. The eightfold way “explains” the laws of quarks, gluons, mesons, etc.
ﬁeld of characteristic p > 0 (algebraically closed) and we want to understand the (polynomial) representations over k. Representations are not semisimple, and the irreducible characters (even their dimensions) are unknown. Symmetric powers of V = kn still make sense. If we let Y1, . . . , Yn be the standard basis of V then Sr V ∼ = homogeneous polynomials of degree r in Y1, . . . , Yn. In fact, Sr V ∼ = k[Y1, . . . , Yn] =: S(V ), the symmetric algebra of V . Let Ip be the ideal of S(V ) generated by the pth powers of the generators Yi . Then Ip ∩ Sr V is a sub-representation of Sr V , and thus Sr V := Sr V /(Ip ∩ Sr V ) is a representation of GLn(k). Call it a truncated symmetric power.
− 1). because Y p i = 0 in the quotient. By tensoring truncated symmetric powers we can get lots of modular representations of GLn(k). For any partition µ = (µ1, . . . , µn) deﬁne: Sµ V := Sµ1 V ⊗ Sµ2 V ⊗ · · · ⊗ Sµn V . This is a truncated generalized symmetric power. Notice that Sµ V is non-zero precisely when µ is n(p − 1)-bounded, meaning that no part µi of µ exceeds n(p − 1). So there are only ﬁnitely many non-zero representations in the family {Sµ V }.
leads to a new symmetric function in Symr that depends on p.An easy calculation shows that hr := character of Sr (V ) = i1≤i2≤···≤ir : deg(xi )<p ∀i xi1 xi2 · · · xir . We call hr a truncated homogeneous symmetric function. If p = 2 then hr = er and if p = ∞ then hr = hr . It follows that if µ = (µ1, µ2, . . . , µn) then hµ := hµ1 hµ2 · · · hµn = character of Sµ (V ). Theorem (D–Walker 1992) Q Symr = Q[h1 , . . . , hn ] and the hr are algebraically independent over Q. This is a p-modular version of the fundamental theorem of symmetric functions. Remark. p does not need to be prime in the above result.
it has a composition series (Jordan–Holder theorem). Hence hµ = character Sµ V = λ kµ λ character L(λ) where L(λ) is the irreducible of highest weight λ and kµ λ counts the number of composition factors isomorphic to L(λ). Here both λ, µ vary over the (ﬁnite) set of n(p − 1)-bounded partitions, so the matrix K = (kµ λ ) is a square matrix of non-negative integers. Conjecture (D 1990) The matrix K is non-singular (i.e., invertible). This is true if p = 2 or n = 2. It has been veriﬁed by computer calculations for a number of other cases, and no counterexample has been found.
. . . , λn) is p-restricted if λi − λi+1 < p for all i. Notice that the set of p-restricted partitions is contained in the set of n(p − 1)-bounded partitions. If the conjecture is true, then by calculating K−1 we could solve for the characters of the L(µ) for all n(p − 1)-bounded µ. By Steinberg’s tensor product theorem this would determine all the irreducible characters of GLn(k). Remark. Around 1979 Lusztig gave a conjecture that would also have determined the irreducible characters of GLn(k). But recently Geordie Williamson found inﬁnitely many counterexamples to Lusztig’s conjecture. (There is now a new version.)
knowing the irreducible characters. However, there is a reformulation of the conjecture that avoids this obstacle. Deﬁnition Let qµ λ (p) be the number of n × n matrices over {0, 1, . . . , p − 1} whose rows sum to µ and columns sum to λ. Then hµ = λ qµ λ (p) mλ (mλ is a monomial symmetric function). So: Theorem (D–Walker 1992) The conjecture is equivalent to the non-singularity of the matrix Q = (qµ λ (p)), rows and columns indexed by the set of n(p − 1)-bounded partitions. This in turn is equivalent to {hµ | µ is n(p − 1)bounded} forming a basis of Q Symn . So calculating the irreducible characters of GLn(k) has been reduced to a purely combinatorial problem about symmetric functions.
1. In my 1982 dissertation, I determined the submodule lattice of symmetric powers Sr V as GLn(k)-module. Len Krop’s 1983 dissertation obtained similar results.In particular, Krop proved Theorem (Krop 1983) The composition factors of Sr V as GLn(k)-module are all of the form Sµ1 V ⊗ (Sµ1 V )(F) ⊗ · · · ⊗ (Sµ1 V )(Fn−1). Here F is the Frobenius pth power map. Ignoring the Frobenius twists in Krop’s theorem, we get the truncated generalized symmetric power Sµ V . 2. Around the same time, the Manchester school of algebraic topologists noticed a strong connection between the modular representation theory of GLn(Fp) and the transfer map in homotopy theory. This connection is eﬀected by commuting actions of GLn(Fp) and the Steenrod algebra on the symmetric algebra S(V ) = Sr V .
GLn(k) where p = the characteristic of k. Notice that GLn(Fp) is a ﬁnite subgroup of GLn(k), and hence any representation of GLn(k) restricts to a representation of GLn(Fp). The modular representation theory of GLn(Fp) is also not semisimple, and we do not know the irreps. . . Thanks for listening!