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POLYNOMIAL BEHAVIOUR of KOSTKA NUMBERS Contents 1 POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS DINUSHI MUNASINGHE + + − Abstract. Given two standard partitions λ+ = (λ1 ≥ · · · ≥ λs ) and λ− = (λ1 ≥ − · · · ≥ λr ) we write λ = (λ+; λ−) and set sλ(x1; : : : ; xt) := sλt (x1; : : : ; xt) where + + − − λt = (λ1 ; : : : ; λs ; 0;:::; 0; −λr ;:::; −λ1 ) has t parts. We provide two proofs that the Kostka numbers kλµ(t) in the expansion X sλ(x1; : : : ; xt) = kλµ(t)mµ(x1; : : : ; xt) µ demonstrate polynomial behaviour, and establish the degrees and leading coefficients of these polynomials. We conclude with a discussion of two families of examples that introduce areas of possible further exploration. Contents 1. Introduction 2 2. Representations of GLn 3 2.1. Representations 3 2.2. Characters of GLn 4 3. Symmetric Polynomials 5 3.1. Kostka Numbers 6 3.2. Littlewood-Richardson Coefficients 6 3.3. Representations and Characters 7 4. Restrictions 7 4.1. Interlacing Partitions 7 4.2. Branching Rules 8 5. Beyond Polynomial Representations 9 6. Weight Decompositions 11 7. Some Notation 11 8. Stability 12 8.1. Kostka Numbers 12 8.2. Littlewood-Richardson Coefficients 13 9. Asymptotics 15 10. The Polynomial Behaviour of Kostka Numbers 16 11. Further Asymptotics 20 1 2 DINUSHI MUNASINGHE 12. Remarks 23 Appendix: Solving Difference Equations 24 References 24 1. Introduction The Kostka numbers, which we will denote kλµ, were introduced by Carl Kostka in his 1882 work relating to symmetric functions. The Littlewood-Richardson coefficients ν cλµ were introduced by Dudley Ernest Littlewood and Archibald Read Richardson in their 1934 paper Group Characters and Algebra. These two combinatorial objects are ubiquitous, appearing in situations in combinatorics and representation theory, which we will address in this work, as well as algebraic geometry. Being such natural objects, the Kostka numbers and Littlewood-Richardson coeffi- cients have been studied extensively. There are explicit combinatorial algorithms for computing both types of coefficients, and many well-established properties. There still exist, however, open questions and areas which remain unexplored. One area of study which contains interesting open questions relating to these coefficients is that of asymp- totics, which we will explore in this work. ν We define the Littlewood-Richardson coefficient cλµ as the multiplicity of the Schur polynomial sν(x1; : : : ; xt) in the decomposition of the product of Schur polynomials X ν sλ(x1; : : : ; xt)sµ(x1; : : : ; xt) = cλµsν(x1; : : : ; xt): ν Similarly, we can define the Kostka number kλµ as the coefficient of mµ(x1; : : : ; xt) in the expansion of the Schur polynomial sλ(x1; : : : ; xt) as X sλ(x1; : : : ; xt) = kλµmµ(x1; : : : ; xt) µ where mµ(x1; : : : ; xt) denotes the minimal symmetric polynomial generated by the mono- mial xµ: In the above formulas, the number of variables, t; may be arbitrary. However, ν cλµ and kλµ become independent of t as it becomes sufficiently large. We begin our study by extending the definition of the Schur polynomial sλ(x1; : : : ; xt) to arbitrary weakly decreasing sequences of integers λ = (λ1 ≥ · · · ≥ λt) by shifting λ to λ + ct; i.e. 1 t sλ(x1; : : : ; xt) = c sλ+c (x1; : : : ; xt): (x1 : : : xt) + + − − Given two standard partitions λ+ = (λ1 ≥ · · · ≥ λs ) and λ− = (λ1 ≥ · · · ≥ λr ) we write λ = (λ+; λ−) and set sλ(x1; : : : ; xt) := sλt (x1; : : : ; xt) + + − − where λt = (λ1 ; : : : ; λs ; 0;:::; 0; −λr ;:::; −λ1 ) has t parts. We consider the expansions X ν (1) sλ(x1; : : : ; xt)sµ(x1; : : : ; xt) = cλµ(t)sν(x1; : : : ; xt) ν POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS 3 and X (2) sλ(x1; : : : ; xt) = kλµ(t)mµ(x1; : : : ; xt) µ where all λ, µ, and ν above are pairs λ = (λ+; λ−); µ = (µ+; µ−), and ν = (ν+; ν−): Both sums in (1) and (2) are finite. A result of Penkov and Styrkas, [PS], implies ν that the coefficients cλµ(t) stabilize for sufficiently large t: The coefficients kλµ(t), on the other hand, demonstrate polynomial behaviour. This result has been known for a while, see for example [BKLS] and the references therein. The methods used by Benkart and collaborators apply to representations of groups beyond GLn; but provide little information about the Kostka polynomials themselves. The purpose of this work is to give two proofs of the fact that kλµ(t) is a polynomial in t: The proofs given are specific to GLn; but will hopefully lead to formulas or algorithms for computing these polynomials explicitly. The first proof uses the stability of the Littlewood-Richardson coefficients, and the second uses a recurrence relation established through branching rules. We end the document with a discussion of two families of examples which will suggest further generalizations and directions of study. Conventions. Unless otherwise stated, we will be working over C; representations are finite dimensional, and N = Z≥0: We denote by [k] the set of integers f1; : : : ; kg: 2. Representations of GLn Definition 1. We define a polynomial representation of the group G = GLn to be a group homomorphism ∼ ρ : G ! GL(V ) = GLN where N = dim(V ) is the dimension of the representation, such that for every A = (aij) 2 GLn, ρ(A) = (ρij(A)), where each entry ρij(A) is a polynomial in the entries akl of A 2 GLn. In this work we will refer to polynomial representations simply as representations of GLn. For a more extensive discussion, one can refer to [GW] and [FH]. 2.1. Representations. All finite dimensional representations of GLn are completely reducible - i.e., if ρ is a representation of GLn such that N = dim(V ) < 1, then there is a decomposition of V as the direct sum of irreducible representations. It is therefore natural to want to parameterize all possible non-isomorphic irreducible representations, and when given a representation, to ask how it could be decomposed as the direct sum of irreducible components. We turn our attention to the Schur-Weyl Duality, which will allow us to build all irreducible representations of GLn. 2.1.1. Schur-Weyl Duality. The Schur-Weyl Duality relates the irreducible representa- tions of the General Linear Group, GLn, to those of the Symmetric group Sk: We begin by considering the kth tensor power of V; V ⊗k, where V = Cn: The General Linear group n ⊗k GLn acts on C by matrix multiplication, so it acts on V by g · (v1 ⊗ · · · ⊗ vk) = g · v1 ⊗ · · · ⊗ g · vk 4 DINUSHI MUNASINGHE ⊗k for g 2 GLn. On the other hand, Sk acts on V by permuting the terms of v1 ⊗· · ·⊗vk 2 V ⊗k. That is to say, σ · (v1 ⊗ · · · ⊗ vn) = vσ−1(1) ⊗ · · · ⊗ vσ−1(n): ⊗k These actions commute, so V is a GLn ×Sk -module. We thus have the decomposition: ⊗k ∼ M λ V = Vλ ⊗ S λ λ where the sum runs over all λ partitioning k, the S are the Sk-modules known as Specht modules, and the Vλ are GLn-modules. In particular, each Vλ is a polynomial GLn-module, and is irreducible if λ has at most n parts and is the zero module otherwise. Furthermore, every irreducible polynomial GLn module is isomorphic to a module Vλ for λ = (λ1 ≥ · · · ≥ λn) with λi 2 Z≥0: Two modules Vλ and Vµ are isomorphic if and only if λ = µ. This gives us a concrete way of realizing the irreducible representations of GLn: We will not prove these results in this work, however proofs can be found in [GW]. Example 1. A simple example is the decomposition of V ⊗2: M V ⊗ V = Sym2V Alt2V where Sym2V is the submodule of V ⊗V generated by all elements of the form u⊗v+v⊗u for u; v 2 V , and Alt2V is the submodule of V ⊗ V generated by the elements of the form u ⊗ v − v ⊗ u for u; v 2 V . While it is clear that V ⊗ V = Sym2V L Alt2V as vector spaces, it is not immediately obvious that this is a decomposition into irreducible submodules. This fact can be established directly. However, it follows cleanly from the Schur-Weyl duality: S2 has two irreducible representations indexed by the corresponding Young tableaux, so we have the decomposition into irreducibles M V ⊗ V = V(2) V(1;1); where S2 acts on V(2) via the trivial representation, and on V(1;1) via the sign represen- tation. Now, V(2) = fx 2 V ⊗ V jσ · x = x; 8σ 2 S2g and V(1;1) = fx 2 V ⊗ V jσ · x = sgn(σ)x; 8σ 2 S2g; 2 2 but these are precisely Sym V and Alt V: Thus, these modules are irreducible GLn modules by the Schur-Weyl duality. 2.2. Characters of GLn. Definition 2. We define the character of a representation ρ of G = GLn to be the map: χV : G ! C such that χV (g) = tr(ρ(g)) for all g 2 G: POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS 5 Since −1 −1 χV (g) = tr(g) = tr(hgh ) = χV (hgh ); the character χV is invariant under conjugation and hence is constant on the conjugacy classes of G: Since the diagonalizable matrices are dense in GLn, the values of χV when restricted to the diagonal matrices completely determine the character χV (since all diagonalizable matrices are, by definition, conjugate to a diagonal matrix, and characters are constant on conjugacy classes). Furthermore, since the n diagonal entries completely determine a diagonal matrix, we can consider the character χV to actually be a map × n χV :(C ) ! C: When we consider the representations Vλ of GLn as discussed in Section 2.1.1, it can be shown that χVλ (x1; : : : ; xn) = sλ(x1; : : : ; xn), i.e. the characters of the representa- tions Vλ of GLn are in fact the Schur polynomials sλ(x1; : : : ; xn): We will discuss these polynomials in the next section.
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