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Local relations between the Jucys-Murphy elements and adjacent transpositions

Statement

Let $n \geq 2$ and put $s_{i} : = (i i + 1)$ for $1 \leq i \leq n - 1$ , the Coxeter generators of $S_{n}$ . Then in $Z [S_{n}]$ , hence in $R [S_{n}]$ for every commutative ring $R$ : (a) $s_{i} X_{j} = X_{j} s_{i}$ whenever $j \notin {i, i + 1}$ ; (b) $s_{i} X_{i} + 1 = X_{i + 1} s_{i}$ ; equivalently $s_{i} X_{i} s_{i} + s_{i} = X_{i + 1}$ and $s_{i} X_{i + 1} = X_{i} s_{i} + 1$ . In particular the subalgebra generated by $s_{i}, X_{i}, X_{i + 1}$ satisfies the local $H (2)$ relations $s_{i}^{2} = 1$ , $X_{i} X_{i + 1} = X_{i + 1} X_{i}$ , $s_{i} X_{i} + 1 = X_{i + 1} s_{i}$ .

Facts & Assumptions

Given: An integer $n \geq 2$ , an index $1 \leq i \leq n - 1$ , the adjacent transposition $s_{i} = (i i + 1)$ , the transposition sums $T_{k} = \sum_{1 \leq p < q \leq k} (p q)$ , and the Jucys-Murphy elements $X_{k} = \sum_{j < k} (j k) = T_{k} - T_{k - 1}$ (The Jucys-Murphy elements of the symmetric group algebra).

[F1]

For $1 \leq k \leq m \leq n$ the element $X_{k}$ of $Z [S_{m}]$ maps to $X_{k}$ under the inclusion $Z [S_{m}] ↪ Z [S_{n}]$ , and each base change $Z [S_{n}] \to R [S_{n}]$ is a unital ring homomorphism carrying $X_{k}$ to $X_{k}$ for $1 \leq k \leq n$ : the coefficient map $\sum_{g} a_{g} g \mapsto \sum_{g} (a_{g} 1_{R}) g$ preserves the group-basis product and the identity (The Jucys-Murphy elements of the symmetric group algebra).

[F2]

The elements $s_{1}, \dots, s_{n - 1}$ generate $S_{n}$ subject to the Coxeter relations; in particular $s_{i}^{2} = 1$ and $s_{i}^{- 1} = s_{i}$ (The symmetric group has the Coxeter presentation).

[F3]

For $h \in S_{n}$ and distinct $a, b \in {1, \dots, n}$ one has $h (a b) h^{- 1} = (h (a) h (b))$ (Conjugating a cycle relabels each entry: $g (a_{1} \dots a_{k}) g^{- 1} = (g (a_{1}) \dots g (a_{k}))$ ).

Proof

technique · direct computation in the group basis

1.1F1F2F3algebra

Case $j < i$ of (a). For such $j$ one has $X_{j} = \sum_{l < j} (l j)$ with $l < j < i$ , and $s_{i}$ fixes both letters $l$ and $j$ ; [F3] and [F2] give $s_{i} (l j) s_{i} = (s_{i} (l) s_{i} (j)) = (l j)$ , hence $s_{i} (l j) = (l j) s_{i}$ . Summing over $l < j$ gives $s_{i} X_{j} = X_{j} s_{i}$ .

1.2F1F2F3algebra

Case $j > i + 1$ of (a). Here $s_{i}$ fixes $j$ , and [F3] gives $s_{i} X_{j} s_{i} = \sum_{l < j} (s_{i} (l) s_{i} (j)) = \sum_{l < j} (s_{i} (l) j)$ . The map $l \mapsto s_{i} (l)$ is a bijection of ${1, \dots, j - 1}$ onto itself, because it interchanges the two elements $i, i + 1$ of that set and fixes all others; hence $\sum_{l < j} (s_{i} (l) j) = \sum_{l < j} (l j) = X_{j}$ and $s_{i} X_{j} = X_{j} s_{i}$ .

1.3F1F2algebra

Part (b). For $j < i$ the two permutations $(i i + 1) (j i)$ and $(j i + 1) (i i + 1)$ agree on $j$ , on $i$ and on $i + 1$ and fix every other letter, hence are equal; and $X_{i + 1} = \sum_{j < i} (j i + 1) + (i i + 1)$ . Using $X_{i} = \sum_{j < i} (j i)$ and $s_{i} = (i i + 1)$ we therefore get $s_{i} X_{i} = \sum_{j < i} (i i + 1) (j i) = \sum_{j < i} (j i + 1) (i i + 1) = (\sum_{j < i} (j i + 1)) s_{i} = (X_{i + 1} - s_{i}) s_{i} = X_{i + 1} s_{i} - s_{i}^{2} = X_{i + 1} s_{i} - 1$ by [F2]. Hence $s_{i} X_{i} + 1 = X_{i + 1} s_{i}$ .

1.4F1F3algebra

The product $X_{i} X_{i + 1}$ . The element $T_{k}$ is central in $Z [S_{k}]$ for every $k$ : by [F3] conjugation by $h \in S_{k}$ sends each transposition $(p q)$ to the transposition $(h (p) h (q))$ , and ${p, q} \mapsto {h (p), h (q)}$ is a bijection of the two-element subsets of ${1, \dots, k}$ , so $h T_{k} h^{- 1} = T_{k}$ . Since $T_{i + 1} = T_{i} + X_{i + 1}$ and $X_{i} \in Z [S_{i}]$ , centrality of $T_{i + 1} \in Z [S_{i + 1}]$ and of $T_{i} \in Z [S_{i}]$ gives $X_{i} X_{i + 1} = X_{i} (T_{i + 1} - T_{i}) = X_{i} T_{i + 1} - X_{i} T_{i} = T_{i + 1} X_{i} - T_{i} X_{i} = (T_{i + 1} - T_{i}) X_{i} = X_{i + 1} X_{i}$ .

2.1step 1.3F2algebra

Equivalent forms. Multiplying $s_{i} X_{i} + 1 = X_{i + 1} s_{i}$ on the right by $s_{i}$ and using $s_{i}^{2} = 1$ from [F2] gives $s_{i} X_{i} s_{i} + s_{i} = X_{i + 1}$ ; multiplying that identity on the left by $s_{i}$ gives $X_{i} s_{i} + 1 = s_{i} X_{i + 1}$ . Thus all three displayed forms of (b) hold.

3.1step 1.1step 1.2step 1.3step 2.1step 1.4F1F2∎

Collecting steps 1.1 and 1.2 covers every $j \notin {i, i + 1}$ , since $j < i$ and $j > i + 1$ are the only possibilities for $1 \leq j \leq n$ ; step 1.3 and step 2.1 give the three listed forms of (b); step 1.4 and [F2] give $X_{i} X_{i + 1} = X_{i + 1} X_{i}$ and $s_{i}^{2} = 1$ . All these identities are equalities in $Z [S_{n}]$ , and by [F1] the base change $Z [S_{n}] \to R [S_{n}]$ carries them to the corresponding identities in $R [S_{n}]$ . Hence (a) and (b) hold, and the subalgebra generated by $s_{i}, X_{i}, X_{i + 1}$ satisfies the three listed $H (2)$ relations.

Remarks

Local meaning of the relations. The identity (b) says that $X_{i + 1}$ is obtained from $X_{i}$ by conjugating with $s_{i}$ and adding $s_{i}$ , so the pair $X_{i}, X_{i + 1}$ together with $s_{i}$ generates a local subalgebra satisfying the $H (2)$ relations: on a joint eigenline of the $X_{k}$ on which $X_{i}$ acts by $a$ and $s_{i}$ by $\pm 1$ , the relation forces $X_{i + 1}$ to act by $a \pm 1$ . This is the computation behind the weight-transposition analysis of The joint spectrum of the Jucys-Murphy elements is the set of tableau content vectors.
What is not claimed. The statement records that the three listed relations hold in the subalgebra generated by $s_{i}, X_{i}, X_{i + 1}$ ; no claim is made that this subalgebra has a presentation with exactly these generators and relations, and no dimension count for it is used on this page.

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Sources

Garsia, Young Seminormal Representation, Murphy Elements and Content Evaluations, UCSD lecture notes (2003), section 3, printed pp. 18-25, Theorem 3.1 (standard reference, not scraped)
Okounkov-Vershik, A New Approach to the Representation Theory of the Symmetric Groups, Selecta Math. (N.S.) 2 (1996) 581-605; complete arXiv repost math/0503040, section 3, printed pp. 12-15 (standard reference, not scraped)
Michael Muger, Tensor Categories: A Selective Guided Tour, section 4 (Coxeter presentation of the symmetric group) (standard reference, not scraped)