2021/10/06 by Darij Grinberg, Grinberg, Darij
Mathematics · #05E05 #15A15 #15A24 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2110.03108
openalex publication_date 2021/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let R be a commutative ring and n≥1 and p≥0 two integers. Let hk, i be an element of R for all k∈\mathbb Z and i∈ [n]. For any α∈\mathbb Zn, we define tα:=det\beginpmatrix hα1+1, 1 amp; hα1+2, 1 amp; ⋯ amp; hα1+n, 1
hα2+1, 2 amp; hα2+2, 2 amp; ⋯ amp; hα2+n, 2
⋮ amp; ⋮ amp; \ddots amp; ⋮
hαn+1, n amp; hαn+2, n amp; ⋯ amp; hαn+n, n \endpmatrix ∈ R (where αi denotes the i-th entry of α). Then, we have the identity ∑_\substackβ∈\0,1,2,…\n ;
|β|=ptα+β =det \beginpmatrix hα1+1, 1 amp; hα1+2, 1 amp; ⋯ amp; hα1+(n-1), 1 amp; hα1+(n+p), 1
hα2+1, 2 amp; hα2+2, 2 amp; ⋯ amp; hα2+(n-1), 2 amp; hα2+(n+p), 2
⋮ amp; ⋮ amp; \ddots amp; ⋮ amp; ⋮
hαn+1, n amp; hαn+2, n amp; ⋯ amp; hαn+(n-1), n amp; hαn+(n+p), n \endpmatrix (where α+β denotes the entrywise sum of the tuples α and β). Furthermore, if p≤ n, then ∑_\substackβ∈\ 0,1\ n ;
| β| =ptα+β=det \beginpmatrix hα1+ξ1 , 1 amp; hα1+ξ2 , 1 amp; ⋯ amp; hα1+ξn , 1
hα2+ξ1 , 2 amp; hα2+ξ2 , 2 amp; ⋯ amp; hα2+ξn , 2
⋮ amp; ⋮ amp; \ddots amp; ⋮
hαn+ξ1 , n amp; hαn+ξ2 , n amp; ⋯ amp; hαn+ξn , n \endpmatrix , where ξ=(1,2,…,n-p,n-p+2,n-p+3,…,n+1). We prove these two identities (in a slightly more general setting, where R is not assumed commutative) and use them to derive some variants of the Pieri rule found in the literature.