## Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers

Venue: | J. Funct. Anal |

Citations: | 12 - 7 self |

### BibTeX

@ARTICLE{Tate_latticepath,

author = {Tatsuya Tate and Steve Zelditch},

title = {Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers},

journal = {J. Funct. Anal},

year = {},

pages = {402--447}

}

### OpenURL

### Abstract

Abstract. We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers V ⊗N λ of an irreducible representation Vλ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope. Contents

### Citations

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Citation Context ...oefficient bN(k) = ( ) N N peaks at the center k = k 2 and by Stirling’s formula r! ∼ √ 2πrr+1 2e−r , bN( N 2 ) ∼ N −1/22N . We measure distance from the center by dN(k) = k − N . We then have 2 (see =-=[F]-=-, Chapter 7 for the first two lines ): ⎧ (CL) C N −1/22N e−2d N (k)2 N , if dN(k) = o(N 2 3) ⎪⎨ bN(k) ∼ (MD) C N −1/22N e −2d N (k)2 N −Nf(2d N (k) √ ) N with f(x) = ∑ ∞ n=2 x 2n (2n)(2n−1) , if dN(k)... |

892 |
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Citation Context ... Racah formula, Littlewood-Richardson rule and others [FH, BD]) rapidly become complicated as the number of factors increases. Our analysis of multiplicities is based on the simple and well-know fact =-=[S]-=- that the multiplicities of lattice paths can be obtained as Fourier coefficients of powers k(w) N of a complex exponential sum of the form k(w) = ∑ c(β)e 〈β,w 〉 , w ∈ C n (1) β∈P with positive coeffi... |

826 |
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Citation Context ...n the exponent correction will be played by the map: 1 µλ : X → Q(λ), µλ(x) := ∑ µ∈Mλ m1(λ; ∑ m1(λ; µ)e µ)e〈 µ,x 〉 〈 µ,x 〉 µ. (11) This map is a homeomorphism from X to the interior of Q(λ) (see e.g. =-=[Fu]-=-), and resembles the moment map of a toric variety, restricted to the real torus in (C ∗ ) m . We define a function δλ on the interior Q(λ) o of the polytope Q(λ) by ( ∑ δλ(x) = log µ∈Mλ µ∈Mλ m1(λ; µ)... |

178 |
The Analysis of Linear Partial
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Citation Context ...S(u) = ∑ u α (31) We apply a steepest descent argument to an integral representation of P c N (γ) (see (36) in Section 1). Our basic reference for the stationary phase for complex phase functions is (=-=[Hö]-=-). α∈S12 TATSUYA TATE AND STEVE ZELDITCH 0.4. Organization. We first prove the results on lattice paths, Theorems 9 and 10, in Section 1. We then deduce the main results on multiplicities, Theorems 4... |

143 | Representations and Invariants of classical groups - Goodman, Wallach - 1998 |

129 |
Representation Theory, Graduate Texts
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(Show Context)
Citation Context ...t α so that g C = t C ⊕ ⊕ We denote by Vλ(µ) the weight space in Vλ for the weight µ. If µ ̸∈ Mλ then Vλ(µ) = 0. Consider the subspace W of Vλ defined by W = ⊕ Vλ(λ + γ) ⊂ Vλ. γ∈Λ ∗ It is well-known (=-=[FH]-=-) that the root space gα maps Vλ(µ) to Vλ(µ + α), and t C maps Vλ(µ) onto itself. Thus, by the decomposition of g C above, the subspace W, which contains the one-dimensional subspace Vλ(λ), is invaria... |

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(Show Context)
Citation Context ...mptotic counting of lattice paths with steps in a convex lattice polytope. Relations between lattice paths and representations have been studied for some time, and one is proved by GrabinerMagyar in (=-=[GM]-=-). We include a proof of an adequate relation for our purposes in Proposition 2.4 (see also Proposition 2.3 for the case of weights). General and conceptually clear relations can be derived from the p... |

17 |
Symplectic Techniques in Physics. Second edition
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Citation Context ... 〉 dm(ξ) = (−2πi) d Vol(O) −1 ∑ w(λ+ρ),H 〉 w∈W sgn(w)ei〈 ∏ , (60) 〈 α, H 〉 O where d = #Φ+ is half the dimension of the coadjoint orbit O = Oλ+ρ, and Vol(O) is the symplectic volume of the orbit. See =-=[GS]-=- for the proof. Thus, by the Weyl character formula, we have ∫ e O i〈 ξ,H 〉 dm(ξ) = (−2πi) d Vol(O) −1 ∆(H/2π) ∏ α∈Φ+ 〈 α, H 〉χλ(H/2π), (61) where ∆ is the Weyl denominator: ∆(H) = ∑ w e2πi〈 wρ,H 〉, a... |

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(Show Context)
Citation Context ...τ = τP(α). Proof. It is straight forward to show that ∂f(τ) = µP(τ) − α, A(τ) = ∂ 2 f(τ). (44) Although one can prove the positivity of the map A(τ) for every τ ∈ X by exactly the same argument as in =-=[SZ]-=-, we give a proof of it for completeness. For each β ∈ S, we set mβ(τ) := c(β)e 〈β,τ 〉/k(τ) so that ∑ β∈S mβ(τ) = 1. We define a probability measure ντ S on X∗ supported on S, depending on τ ∈ X, by d... |

13 |
The path model, the quantum Frobenius map and Standard Monomial Theory, “Algebraic Groups and Their Representations
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Citation Context ... our purposes in Proposition 2.4 (see also Proposition 2.3 for the case of weights). General and conceptually clear relations can be derived from the path discussed in Littelmann’s expository article =-=[Lit]-=-. We add some further comments in Section 4. Let us now recall what the combinatorics of lattice paths is about: Given a set S ⊂ N m of allowed steps, an S- lattice path of length N from 0 to β is a s... |

9 | The theory of large deviations: from Boltzmann’s 1877 calculation to equilibrium macrostates
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(Show Context)
Citation Context ...ails of the asymptotics. To introduce our results, we recall one of the first and most basic results of a similar kind, namely Boltzmann’s analysis of the asymptotics of multinomial coefficients (see =-=[E]-=- for historical background and the relation to the present problem): ⎧ ⎨ mN : {(k1, . . .,km) ∈ Nm : k1 + · · · + km ≤ N} → R +, ⎩ mN(k1, . . .,kn) = ( N k1,...,km ) N! = k1!···km! .LATTICE PATH COMB... |

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4 |
A large deviation principle for the reduction of product representations
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(Show Context)
Citation Context ...e function Iλ(x) given by (5). The lower bound will follow from our pointwise asymptotics. We should note the large deviations principle with the rate function (5) has already been proved by Duffield =-=[D]-=- for dMλ,N by a different method. These results give the bulk properties of the measures dmλ,N, dMλ,N in that they give the exponents of the measures of N-independent closed/open sets. Our main result... |

4 |
Harmonic analysis on toric varieties
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(Show Context)
Citation Context ...n the connections between lattice paths and weight multiplicites and on the symplectic model for tensor product multiplicities. Acknowledgments This paper originated as a by-product of our joint work =-=[TSZ1]-=- with B. Shiffman on the Szegö kernel of a toric variety and its applications, where the partition function of a certain lattice path problem appeared. The probabilistic background became clearer in d... |

3 | Estimation asymptotique des multiplicités dans les puissances tensorielles d’un g-module - Biane - 1993 |

2 | Projections of orbits and asymptotics behavior of multiplicities for compact connected Lie groups - Heckmann - 1982 |