A minimal Set of Generators for the Ring of multisymmetric Functions

Rydh, David. "A minimal Set of Generators for the Ring of multisymmetric Functions." Annales de l’institut Fourier 57.6 (2007): 1741-1769. <http://eudml.org/doc/10276>.

@article{Rydh2007,
abstract = {The purpose of this article is to give, for any (commutative) ring $A$, an explicit minimal set of generators for the ring of multisymmetric functions $\{\mathrm\{T\}S\}^d_A(A[x_1,\dots ,x_r])= \bigl (A[x_1,\dots ,x_r]^\{\otimes _A d\}\bigr )^\{\{\mathfrak\{S\}\}_d\}$ as an $A$-algebra. In characteristic zero, i.e. when $A$ is a $\{\mathbb\{Q\}\}$-algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously obtained by Fleischmann.As $\Gamma ^d_A(A[x_1,\dots ,x_r])=\{\mathrm\{T\}S\}^d_A(A[x_1,\dots ,x_r])$ we also obtain generators for divided powers algebras: If $B$ is a finitely generated $A$-algebra with a given surjection $A[x_1,x_2,\dots ,x_r]\rightarrow B$ then using the corresponding surjection $\Gamma ^d_A(A[x_1,\dots ,x_r])\rightarrow \Gamma ^d_A(B)$ we get generators for $\Gamma ^d_A(B)$.},
affiliation = {KTH Department of Mathematics 100 44 Stockholm (Sweden)},
author = {Rydh, David},
journal = {Annales de l’institut Fourier},
keywords = {Symmetric functions; generators; divided powers; vector invariants; multisymmetric functions},
language = {eng},
number = {6},
pages = {1741-1769},
publisher = {Association des Annales de l’institut Fourier},
title = {A minimal Set of Generators for the Ring of multisymmetric Functions},
url = {http://eudml.org/doc/10276},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Rydh, David
TI - A minimal Set of Generators for the Ring of multisymmetric Functions
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 6
SP - 1741
EP - 1769
AB - The purpose of this article is to give, for any (commutative) ring $A$, an explicit minimal set of generators for the ring of multisymmetric functions ${\mathrm{T}S}^d_A(A[x_1,\dots ,x_r])= \bigl (A[x_1,\dots ,x_r]^{\otimes _A d}\bigr )^{{\mathfrak{S}}_d}$ as an $A$-algebra. In characteristic zero, i.e. when $A$ is a ${\mathbb{Q}}$-algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously obtained by Fleischmann.As $\Gamma ^d_A(A[x_1,\dots ,x_r])={\mathrm{T}S}^d_A(A[x_1,\dots ,x_r])$ we also obtain generators for divided powers algebras: If $B$ is a finitely generated $A$-algebra with a given surjection $A[x_1,x_2,\dots ,x_r]\rightarrow B$ then using the corresponding surjection $\Gamma ^d_A(A[x_1,\dots ,x_r])\rightarrow \Gamma ^d_A(B)$ we get generators for $\Gamma ^d_A(B)$.
LA - eng
KW - Symmetric functions; generators; divided powers; vector invariants; multisymmetric functions
UR - http://eudml.org/doc/10276
ER -