aa r X i v : . [ m a t h . C O ] M a y On exponential growth of degrees
Yuval Roichman ∗ August 16, 2018
Abstract
A short proof to a recent theorem of Giambruno and Mishchenko is given in this note.
The following theorem was recently proved by Giambruno and Mishchenko.
Theorem 1.1. [1, Theorem 1] For every < α < , there exist β > and n ∈ N , such that forevery partition λ of n ≥ n with max { λ , λ ′ } < αnf λ ≥ β n . The proof of Giambruno and Mishchenko is rather complicated and applies a clever order onthe cells of the Young diagram. It should be noted that Theorem 1.1 is an immediate consequenceof Rasala’s lower bounds on minimal degrees [2, Theorems F and H]. The proof of Rasala is verydifferent and not less complicated; it relies heavily on his theory of degree polynomials. In thisshort note we suggest a short and relatively simple proof to Theorem 1.1.First, note that the following weak version is an immediate consequence of the hook-lengthformula.
Lemma 1.2.
The theorem holds for every < α < e .Proof. Under the assumption, for every ( i, j ) ∈ [ λ ] h i,j ≤ h , ≤ λ + λ ′ ≤ αn. Hence, by the hook formula together with Stirling formula, for sufficiently large nf λ = n ! Q ( i,j ) ∈ [ λ ] h i,j ≥ n !(2 αn ) n ≥ ( ne ) n (2 αn ) n = β n , where, by assumption, β := eα > ∗ Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel. [email protected] . Two lemmas
Lemma 2.1.
For every λ ⊢ n Y ( i,j ) ∈ [ λ ]1
For λ = ( λ , λ , . . . , λ t ) ⊢ n let ¯ λ := ( λ , . . . , λ t ) ⊢ n − λ . Then1 ≤ f ¯ λ = ( n − λ )! Q ( i,j ) ∈ [ λ ]1
For every λ ⊢ n and ≤ k ≤ λ Y (1 ,j ) ∈ [ λ ] h j ≤ (cid:18) nk (cid:19) ( λ + ⌊ n − λ k ⌋ )! . Proof.
Obviously, h , > h , > · · · > h ,λ . Since h , ≤ n it follows that Y (1 ,j ) ∈ [ λ ] j ≤ k h j ≤ ( n ) k and Y (1 ,j ) ∈ [ λ ] k For the sake of simplicity the floor notation is omitted in this section.By Lemmas 2.1 and 2.2, f λ = n ! Q ( i,j ) ∈ [ λ ] h ij = n ! Q (1 ,j ) ∈ [ λ ] h j Q ( i,j ) ∈ [ λ ]1
1) it follows that f ( γ n ) ≤ f ( α ) if 1 − e ≤ α and ≤ f ( e ) otherwise. Choosing ǫ = ǫ ( α ) such that ǫ ≤ δ min { − α, e } for some very small δ > n →∞ inf( f λ ) /n ≥ min γ ∈ [ e ,α ] ǫ ǫ (1 − ǫ ) − ǫ γ γ (1 − γ ) − γ ≥ min { f ( δ e ) f ( e ) , f ( δ (1 − α )) f (1 − α ) } > , completing the proof. Acknowledgements. Thanks to Amitai Regev for fruitful discussions and references. References [1] A. Giambruno and S. Mishchenko, Irreducible characters of the symmetric group and exponentialgrowth , arXiv:1406.1653 .[2] R. Rasala, On the minimal degrees of characters of S n , J. Algebra45