The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913711192539136 |
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| author | Nathanson, Melvyn B. |
| author_facet | Nathanson, Melvyn B. |
| contents | Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbf{R}^n$ with nonnegative coordinates. Permuting the coordinates, we can assume that $a_1 \geq \cdots \geq a_n$ and $b_1 \geq \cdots \geq b_n$. The vector $\mathbf{a}$ majorizes the vector $\mathbf{b}$, denoted $\mathbf{b} \preceq \mathbf{a}$, if $\sum_{i=1}^n b_i = \sum_{i=1}^n a_i$ and $\sum_{i=1}^k b_i \leq \sum_{i=1}^k a_i$ for all $k \in \{1,\ldots,n-1\}$. This paper proves theorems of Hardy-Littlewood-Pólya and Rado that $\mathbf{b} \preceq \mathbf{a}$ if and only if $P\mathbf{a} = \mathbf{b}$ for some doubly stochastic matrix $P$ if and only if $\mathbf{b}$ is in the $S_n$-permutohedron generated by $\mathbf{a}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_01746 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron Nathanson, Melvyn B. Combinatorics 05E05, 11B83, 15B51, 26D05, 26D15, 52A20, 52A30, 52A40 Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbf{R}^n$ with nonnegative coordinates. Permuting the coordinates, we can assume that $a_1 \geq \cdots \geq a_n$ and $b_1 \geq \cdots \geq b_n$. The vector $\mathbf{a}$ majorizes the vector $\mathbf{b}$, denoted $\mathbf{b} \preceq \mathbf{a}$, if $\sum_{i=1}^n b_i = \sum_{i=1}^n a_i$ and $\sum_{i=1}^k b_i \leq \sum_{i=1}^k a_i$ for all $k \in \{1,\ldots,n-1\}$. This paper proves theorems of Hardy-Littlewood-Pólya and Rado that $\mathbf{b} \preceq \mathbf{a}$ if and only if $P\mathbf{a} = \mathbf{b}$ for some doubly stochastic matrix $P$ if and only if $\mathbf{b}$ is in the $S_n$-permutohedron generated by $\mathbf{a}$. |
| title | The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron |
| topic | Combinatorics 05E05, 11B83, 15B51, 26D05, 26D15, 52A20, 52A30, 52A40 |
| url | https://arxiv.org/abs/2109.01746 |