RECORD
Multiplication of polynomials on Hermitian symmetric spaces and Littlewood-Richardson coefficients
Representation Theory · Submitted 26 May 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity free. If V_{\lambda} is an irreducible representation of K, let S_{\lambda} denote the corresponding isotypic component of S. Write S_{\lambda} S_{\mu} for the subspace of S spanned by products of S_{\lambda} and S_{\mu}. If V_{\nu} occurs as an irreducible constituent of the tensor product of V_{\lambda} and V_{\mu}, is it true that S_{\nu} is contained in S_{\lambda} S_{\mu}? We investigate this question for representations arising in the context of Hermitian symmetric pairs. We show that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. We also show how the conjecture connects multiplication in the ring S to the usual Littlewood-Richardson rule.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let K be a complex reductive algebraic group and V a representation of K. | background 1.00 | 0.05 | 0.01 | asserted 0.52 |
| s1 | Let S denote the ring of polynomials on V. | background 0.99 | 0.11 | 0.01 | asserted 0.73 |
| s2 | Assume that the action of K on S is multiplicity free. | background 0.96 | 0.07 | 0.02 | asserted 0.90 |
| s3 | If V_{\lambda} is an irreducible representation of K, let S_{\lambda} denote the corresponding isotypic component of S. | background 0.99 | 0.09 | 0.01 | asserted 0.69 |
| s4 | Write S_{\lambda} S_{\mu} for the subspace of S spanned by products of S_{\lambda} and S_{\mu}. | background 0.95 | 0.23 | 0.02 | asserted 0.75 |
| s5 | If V_{\nu} occurs as an irreducible constituent of the tensor product of V_{\lambda} and V_{\mu}, is it true that S_{\nu} is contained in S_{\lambda} S_{\mu}? | background 0.66 | 0.05 | 0.03 | unclear 0.73 |
| s6 | We investigate this question for representations arising in the context of Hermitian symmetric pairs. | contribution 0.86 | 0.64 | 0.03 | asserted 0.59 |
| s7 | We show that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. | result 0.98 | 0.89 | 0.88 | asserted 0.98 |
| s8 | We also show how the conjecture connects multiplication in the ring S to the usual Littlewood-Richardson rule. | contribution 0.51 | 0.90 | 0.09 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity free. If V_{\lambda} is an irreducible representation of K, let S_{\lambda} denote the corresponding isotypic component of S. Write S_{\lambda} S_{\mu} for the subspace of S spanned by products of S_{\lambda} and S_{\mu}. If V_{\nu} occurs as an irreducible constituent of the tensor product of V_{\lambda} and V_{\mu}, is it true that S_{\nu} is contained in S_{\lambda} S_{\mu}? We investigate this question for representations arising in the context of Hermitian symmetric pairs. We show that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. We also show how the conjecture connects multiplication in the ring S to the usual Littlewood-Richardson rule.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.