On Exponential Growth of Degrees

A short proof to a recent theorem of Giambruno and Mishchenko is given in this note. 1 The theorem The following theorem was recently proved by Giambruno and Mishchenko. Theorem 1. [1, Theorem 1] For every 0 < α < 1, there exist β > 1 and n0 ∈ N, such that for every partition λ of n > n0 with max{λ1, λ1} < αn f > β. The proof of Giambruno and Mishchenko is rather complicated and applies a clever order on the cells of the Young diagram. It should be noted that Theorem 1 is an immediate consequence of Rasala’s lower bounds on minimal degrees [2, Theorems F and H]. The proof of Rasala is very different and not less complicated; it relies heavily on his theory of degree polynomials. In this short note we suggest a short and relatively simple proof to Theorem 1. First, note that the following weak version is an immediate consequence of the hooklength formula. Lemma 2. The theorem holds for every 0 < α < 1 2e . Proof. Under the assumption, for every (i, j) ∈ [λ] hi,j 6 h1,1 6 λ1 + λ ′ 1 6 2αn. the electronic journal of combinatorics 22(2) (2015), #P2.58 1 Hence, by the hook-length formula together with Stirling’s formula, for sufficiently large n f = n! ∏ (i,j)∈[λ] hi,j > n! (2αn)n > ( e ) (2αn)n = β, where, by assumption, β := 1 2eα > 1. 2 Two lemmas Lemma 3. For every λ ` n ∏ (i,j)∈[λ] 1 h1,2 > · · · > h1,λ1 . Since h1,1 6 n it follows that ∏ (1,j)∈[λ] j6k h1j 6 (n)k