On some combinatorial aspects of Representation Theory

On some combinatorial aspects of Representation Theory Doctoral Defense Waldeck Schützer schutzer@math.rutgers.edu Rutgers University March 24, 2004 Representation Theory and Combinatorics p.1/46

Overview Degree one nonsymmetric Pieri rule New recursions for characters and multiplicities Complexity of character formulas Representation Theory and Combinatorics p.2/46

Part 1 Degree 1 nonsymmetric Pieri rule Representation Theory and Combinatorics p.3/46

The classical Pieri rule The classical Pieri rule is a combinatorial description for the product of a Schur polynomial with a complete (or an elementary) symmetric polynomial: h m s µ = λ µ hor. m strip s λ. This is a special case of the well-known Littlewood-Richardson rule for the product of any two Schur polynomials. Representation Theory and Combinatorics p.4/46

Jack polynomials Jack polynomials J (α) λ polynomials. generalize Schur In fact, J (1) λ is a scalar multiple of s λ. The Pieri rule for these was found by Stanley in 1989: J (α) m J (α) µ = λ µ hor. m strip c λ m,µ(α)j (α) λ. The Littlewood-Richardson rule is not known for the J (α) λ. Representation Theory and Combinatorics p.5/46

The nonsymmetric Pieri rule Nonsymmetric analogs F η (α) where introduced in 1995 by Heckman and Opdam. They are indexed by compositions, i.e. η Z n +. The Pieri rule for them is F ν (α) F η (α) = λ g λ νη(α)f (α) λ where ν {0, 1} n. This is not fully known yet, but we will give a complete answer for the case ν = ε k. Representation Theory and Combinatorics p.6/46

Nonsymmetric Jack polynomials Up to a scalar multiple, F η can be defined as a simultaneous polynomial eigenfunction for the commuting family of Cherednik operators ξ i = αx i + x i ki x k x i x k (1 s ik ) + 1 i, for i = 1,..., n, with eigenvalues η i = η i α #{k i η k > η i }. Here s ik is the transposition permuting x i and x k. Representation Theory and Combinatorics p.7/46

Main Results Two main questions answered for the product F εk F η : Which polynomials occur in the decomposition? What are values of the coefficients? To answer the first question we need a few definitions. Representation Theory and Combinatorics p.8/46

Composition diagrams The diagram of η is the set diag(η) = {(i, j) 1 i n, 1 j η i }. Representation Theory and Combinatorics p.9/46

Composition diagrams The diagram of η is the set diag(η) = {(i, j) 1 i n, 1 j η i }. Graphically, for η = (312): 3 3 3 3 j or better 3 3 i Representation Theory and Combinatorics p.9/46

Composition diagrams The diagram of η is the set diag(η) = {(i, j) 1 i n, 1 j η i }. Graphically, for η = (312): 3 3 3 3 j or better 3 3 i Usually we identify diag(η) with η. Representation Theory and Combinatorics p.9/46

An ordering on compositions Define an ordering on compositions by asking that ν η if there exists a permutation π such that ν i < η π(i), if i < π(i), ν i η π(i), if i π(i). This extends the usual ordering (inclusion of diagrams). Representation Theory and Combinatorics p.10/46

Minimal elements above η For L = {j 1,..., j l } {1,..., n}, let η = (η 1,..., η j1,..., η j2,..., η jl,..., η n ), and define µ = C L (η) by µ = (η 1,..., η j2,..., η j3,..., η j1 + 1,..., η n ). η = j 1 j 2 j 3 c L (η) = Representation Theory and Combinatorics p.11/46

First main theorem Theorem. gε λ k η 0 if and only if: (i) λ = C S (η) where S = {i 1,..., i s } := {i η i λ i }, (ii) Either i s k or #{i k η i = η i1 + 1} > 0, (iii) if η i1 = η 1 = = η k then i 1 < k. k k Representation Theory and Combinatorics p.12/46

Example The following are all compositions occurring in the expansion of F ε2 F (1121) : = These are some compositions which do not occur for the indicated reason: k Not minimal Wrong i 1 Representation Theory and Combinatorics p.13/46

Remarks S need not be the only set to satisfy (i). In fact, as we will see below, there is always a (unique) maximal set L such that λ = c L (η). Representation Theory and Combinatorics p.14/46

Maximal sets A subset L = {j 1,..., j l } {1,..., n} is maximal w.r.t. η if: η i η j1, if i < j 1, η i η jt, if j t 1 j l. One can show that if λ = C M (η) = C L (η) and L is maximal, then M L, hence λ = C L (η) determines L uniquely. In particular, S L. Representation Theory and Combinatorics p.15/46

Remarks Let S L. Then (ii) is equivalent to j l k. We are going to describe g λ ε k η combinatorially. We will need two auxiliary polynomials b ηλ (α) and b (k) ηλ (α). They will be defined by playing a jeu de flèches (an arrow shooting game). First, we recall some fundamental concepts. Representation Theory and Combinatorics p.16/46

Composition statistics For a box s = (i, j) of η we define the coarm length and the arm lengths as so coarm a η(s) s arm a η (s) # = a η(s) = j 1, # = a η (s) = η i j. Representation Theory and Combinatorics p.17/46

Composition statistics For a box s = (i, j) of η we define the leg length as : so # = l η (s) = #{k i j η k η i }. Representation Theory and Combinatorics p.17/46

Composition statistics For a box s = (i, j) of η we define the coleg length as : so # = l η(s) = #{k i η k > η i }. Representation Theory and Combinatorics p.17/46

Composition statistics For a box s = (i, j) of η we define the lower hook length polynomial as d η(s) = (a η (s) + 1)α + l η (s). : e.g. d η(3, 2) = 2α + 3. We also define the upper hook polynomial as d η (s) = d η(s) + 1. Representation Theory and Combinatorics p.17/46

Composition statistics The content of a box in η is defined as c η (s) = (a η(s) + 1)α l η(s). : e.g. c η (3, 2) = 2α 1. Remark: the eigenvalue η i is just the content c η (i, η i ) of the rightmost box on row i of η. Representation Theory and Combinatorics p.17/46

Hook tableaux Let η = (01312), λ = (13211) and k = 1. Then L = {1, 2, 3, 5} is maximal w.r.t. η and λ = C L (η). α+2 d λ α+3 d η 3α+4 2α+2 α 3α+5 2α+3 α+1 2α+3 α+1 α+1 d η α+2 d λ 2α+4 α+3 α Representation Theory and Combinatorics p.18/46

Hook tableaux Let η = (01312), λ = (13211) and k = 1. Then L = {1, 2, 3, 5} is maximal w.r.t. η and λ = C L (η). For the moment, forget the non-colored rows and think of the colored ones as if they formed a cycle, namely regard the first row as if it came below the last one. Place the arrows close to the right end of each (colored) row as follows. Representation Theory and Combinatorics p.18/46

Hook tableaux Let η = (01312), λ = (13211) and k = 1. Then L = {1, 2, 3, 5} is maximal w.r.t. η and λ = C L (η). α+3 3α+5 2α+3 α+1 α+2 3α+4 2α+2 α 2α+3 α+1 α+1 α+2 2α+4 α+3 α The arrows can only reach as far as the next row in cycle. Representation Theory and Combinatorics p.18/46

Hook tableaux Let η = (01312), λ = (13211) and k = 1. Then L = {1, 2, 3, 5} is maximal w.r.t. η and λ = C L (η). 1 α+3 3α+5 2α+3 1 3α+4 1 α 2α+3 α+1 α+1 α+2 2α+4 1 α b ηλ :=2 (α+3)(3 α+5)(2 α+3) 2 (α+1) 2 (α+2) 2 (3 α+4)α 2 Representation Theory and Combinatorics p.18/46

Hook tableaux Let η = (01312), λ = (13211) and k = 1. Then L = {1, 2, 3, 5} is maximal w.r.t. η and λ = C L (η). Now we are going to take k into account. Do so by placing a shield on row k protecting from the arrows. Then repeat the jeu de flèches. Representation Theory and Combinatorics p.18/46

Hook tableaux Let η = (01312), λ = (13211) and k = 1. Then L = {1, 2, 3, 5} is maximal w.r.t. η and λ = C L (η). α+2 shield α+3 3α+4 1 α 3α+5 2α+3 1 2α+3 α+1 α+1 α+2 2α+4 1 α b (1) ηλ := 2 (α+3)(3 α+5)(2 α+3)2 (α+1) 2 (α+2) 3 (3 α+4)α 2 Representation Theory and Combinatorics p.18/46

Second main theorem Theorem. If gε λ k η 0 then (α + k)b (k) ηλ + (c λ(j p ) c λ (j l ))b ηλ, gε λ if k = j k η = p L, (c λ (j p ) c λ (j l ))b ηλ, if j p < k < j p+1, αb ηλ, if k < j 1, where c λ (i) := c λ (i, λ i ) = λ i α l λ(i) is the content of the rightmost box on row i of λ. Representation Theory and Combinatorics p.19/46

Example Using the aforementioned values of b ηλ and b (1) ηλ for η = (01312) and λ = (13211), we find that and thus c λ (1) = α 2, c λ (5) = α 4, g λ 1η = (α+1)b (1) ηλ +((α 2) (α 4))b ηλ = 2(α+1) 2 (α+3) 2 (3α+5)(2α+3) 2 (α+2) 2 (3α+4)α 3 Representation Theory and Combinatorics p.20/46

Example For η = (045), λ = (451) and k = 2, we find that g (451) 2,(045) = 24(α + 2)(2α + 1) 3 (3α + 1) 2 (α + 1) 2 (5α + 3)(3α + 2) (4α + 1)(5α + 2)(α 1)α 5. Thus it is not always possible to define a global sign for each coefficient. Representation Theory and Combinatorics p.21/46

Idea of the proof There are some recursions for the g λ ε k η. There is an exact, non-combinatorial formula for x i F η by Marshall. First guess which λ will occur and guess the value of g λ ε k η. Then use either recursion or closed formula to show the guess is correct. End of part 1 Representation Theory and Combinatorics p.22/46

Part 2 New recursions for characters and multiplicities Representation Theory and Combinatorics p.23/46

Preliminaries - 1 g - simple Lie algebra of rank n Representation Theory and Combinatorics p.24/46

Preliminaries - 1 g - simple Lie algebra of rank n h g - Cartan subalgebra Representation Theory and Combinatorics p.24/46

Preliminaries - 1 g - simple Lie algebra of rank n h g - Cartan subalgebra ω 1,..., ω n h - fundamental weights Representation Theory and Combinatorics p.24/46

Preliminaries - 1 g - simple Lie algebra of rank n h g - Cartan subalgebra ω 1,..., ω n h - fundamental weights P = Z ω 1,..., ω n - weight lattice Representation Theory and Combinatorics p.24/46

Preliminaries - 1 g - simple Lie algebra of rank n h g - Cartan subalgebra ω 1,..., ω n h - fundamental weights P = Z ω 1,..., ω n - weight lattice P + = Z + ω 1,..., ω n - the dominant weights Representation Theory and Combinatorics p.24/46

Preliminaries - 1 g - simple Lie algebra of rank n h g - Cartan subalgebra ω 1,..., ω n h - fundamental weights P = Z ω 1,..., ω n - weight lattice P + = Z + ω 1,..., ω n - the dominant weights = {α 1,..., α n } P - simple roots Representation Theory and Combinatorics p.24/46

Preliminaries - 2 α i = 2α i α i,α i - the coroots Representation Theory and Combinatorics p.25/46

Preliminaries - 2 α i = 2α i α i,α i - the coroots µ is regular whenever µ, α > 0 for all α. Representation Theory and Combinatorics p.25/46

Preliminaries - 2 α i = 2α i α i,α i - the coroots µ is regular whenever µ, α > 0 for all α. W - the Weyl group w/ generators s i := s αi Representation Theory and Combinatorics p.25/46

Preliminaries - 2 α i = 2α i α i,α i - the coroots µ is regular whenever µ, α > 0 for all α. W - the Weyl group w/ generators s i := s αi R = W {α 1,..., α n } - the root system Representation Theory and Combinatorics p.25/46

Preliminaries - 2 α i = 2α i α i,α i - the coroots µ is regular whenever µ, α > 0 for all α. W - the Weyl group w/ generators s i := s αi R = W {α 1,..., α n } - the root system R + - positive roots Representation Theory and Combinatorics p.25/46

Preliminaries - 2 α i = 2α i α i,α i - the coroots µ is regular whenever µ, α > 0 for all α. W - the Weyl group w/ generators s i := s αi R = W {α 1,..., α n } - the root system R + - positive roots V λ - Irreducible f.d. g-module of highest weight λ Representation Theory and Combinatorics p.25/46

The character The (formal) character of V λ is χ λ = µ P λ dim(v λ (µ))e µ, where V λ (µ) = {v V λ h v = µ(h)v, h h} is the weight space of V λ of weight µ relative to h. This makes sense since h is abelian and consists of semisimple elements. Representation Theory and Combinatorics p.26/46

The WCF The Weyl Character Formula (WCF) states that δχ λ = w W ε(w)e w(λ+ρ) where ε(w) = ( 1) l(w) and δ is the Weyl denominator δ = ε(w)e wρ = (e α/2 e α/2 ). w W α R + Representation Theory and Combinatorics p.27/46

The girdle Is defined as θ λ = µ P λ e µ. Looks like the character, except the coefficients are all 1 s. Representation Theory and Combinatorics p.28/46

A partition function Let R + = R + \ be the positive non-simple roots. For any φ R +, let φ = β R + β Then we can write where α R +(1 e α ) = 1 + φ c φ e φ, c β = #{ φ = β #φ is even} #{ φ = β #φ is odd}. Representation Theory and Combinatorics p.29/46

Signed characters Above we defined χ λ when λ is dominant. Representation Theory and Combinatorics p.30/46

Signed characters Above we defined χ λ when λ is dominant. For any µ P, let w µ W be minimal s.t. w µ (µ + ρ) is dominant. Representation Theory and Combinatorics p.30/46

Signed characters Above we defined χ λ when λ is dominant. For any µ P, let w µ W be minimal s.t. w µ (µ + ρ) is dominant. Set µ = w µ (µ + ρ) ρ. Note µ is dominant when µ + ρ is regular. Representation Theory and Combinatorics p.30/46

Signed characters Above we defined χ λ when λ is dominant. For any µ P, let w µ W be minimal s.t. w µ (µ + ρ) is dominant. Set µ = w µ (µ + ρ) ρ. Note µ is dominant when µ + ρ is regular. Let ε µ = ( 1) l(w µ), if µ + ρ is regular, and zero otherwise. Representation Theory and Combinatorics p.30/46

Main Theorem Theorem. For a dominant λ, Θ λ = χ λ + φ c φ χ λ φ is a recursion for χ λ. Representation Theory and Combinatorics p.31/46

Idea of the proof We show that both sides of Θ λ = χ λ + φ c φ χ λ φ are equal to w W ε(w) e wλ α R +(1 e wα ). Representation Theory and Combinatorics p.32/46

LHS Proposition. Θ λ = w W ε(w) e wλ α R +(1 e wα ). We give an elementary proof, but this also follows from the work of Brion. Representation Theory and Combinatorics p.33/46

RHS Proposition. w W ε(w) e wλ α R +(1 e wα ) = χ λ + φ c φ χ λ φ. The proof is: just expand and use the WCF. Representation Theory and Combinatorics p.34/46

Example For g of type A 2, χ λ = Θ λ + χ λ ρ. For g of type B 2, χ λ = Θ λ + χ λ α1 α 2 + χ λ α1 2α 2 χ λ 2α1 3α 2 Representation Theory and Combinatorics p.35/46

Consequences Corollary. m λ (µ) = 1 φ c φ m λ φ (µ), where m λ (µ) = ε λ dim V λ (µ). Representation Theory and Combinatorics p.36/46

Consequences Corollary. m λ (µ) = 1 φ c φ m λ φ (µ), where m λ (µ) = ε λ dim V λ (µ). Corollary. P λ = dim V λ + φ ε λ φ c φ dim V λ φ. Representation Theory and Combinatorics p.36/46

Consequences Corollary. P λ = w W 1 n i=1 ρ, wα i n j=0 ρ, wλ j j! T n j ( wα 1,..., wα n ), where T i is the i-th Todd polynomial defined by i 1 tx i 1 e tx i = i=0 T (x 1, x 2,...)t i. Representation Theory and Combinatorics p.37/46

Example For g of type A 2, and λ = (a, b), we get ( ) a + b + 2 P λ = + ab. 2 Representation Theory and Combinatorics p.38/46

Example For g of type B 2, m (i,j) (0,0) = 0 unless j is even. So when λ = (i, 2j) for i, j Z + we have the recursion m λ (0,0)=1+m λ α1 α 2 (0,0)+m λ α1 2α 2 (0,0) m λ 2α1 3α 2 (0,0). This can be solved exactly: m (i,2j) (0,0) = 1 + ( 1)i 4 + End of Part 2 (i + 1)(2j + 1) 2. Representation Theory and Combinatorics p.39/46

Part 3 The complexity of character formulas Representation Theory and Combinatorics p.40/46

Two character formulas We studied two character formulas: One classic: Freudenthal s formula expressing the multiplicity of a weight in terms of multiplicity for higher weights. One new: Sahi s formula is a recursion for non-symmetric analogs P λ of characters χ λ. Representation Theory and Combinatorics p.41/46

Statement of the Problem Given a dominant weight λ of length m, we want to compute the character table T m, containing the characters for all dominant weights µ whose length is m. Representation Theory and Combinatorics p.42/46

The size of the Problem The size of χ λ is precisely P λ and this is about O(m n ). Thus the size of T m is about O(m 2n ). Representation Theory and Combinatorics p.43/46

Computational model We adopted straight-line program model with uniform cost function, so that the time complexity depends only on the total number of arithmetic operations. Hence, an optimal method for computing T m, takes O(m 2n ) time. Representation Theory and Combinatorics p.44/46

Main Theorem Theorem. Let λ be a weight of length m. Then Freudenthal computes T m in O(m 2n+1 ) time while Sahi computes it in O(m 2n ) time, therefore Sahi is of optimal performance. Representation Theory and Combinatorics p.45/46

Computational experience Representation Theory and Combinatorics p.46/46

End of Part 3 Representation Theory and Combinatorics p.47/46