LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Degenerate affine Hecke ordered monomials form a basis

Statement

Let $F$ be any field and let $H_{d}^{\deg}$ be the algebra generated by commuting $x_{1}, \dots, x_{d}$ and symmetric-group generators $s_{1}, \dots, s_{d - 1}$ with $s_{r} x_{r} = x_{r + 1} s_{r} - 1, s_{r} x_{r + 1} = x_{r} s_{r} + 1, s_{r} x_{j} = x_{j} s_{r} (j \notin {r, r + 1}),$ together with the symmetric-group quadratic, braid and distant commutation relations, as in the degenerate clause of Affine type-A presentations and charged cyclotomic quotients (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation, The symmetric group has the Coxeter presentation). Then ${x_{1}^{a_{1}} \dots x_{d}^{a_{d}} w : a_{j} \geq 0, w \in S_{d}}$ is an $F$ -basis of $H_{d}^{\deg}$ . No cyclotomic basis or Brundan–Kleshchev isomorphism is assumed. For $d = 0$ the basis is the singleton empty monomial.

Facts & Assumptions

Given: A field $F$ , the polynomial ring $P = F [z_{1}, \dots, z_{d}]$ , its fraction field $K = F (z_{1}, \dots, z_{d})$ , and the crossed product $K ⋊ S_{d}$ with basis the group elements and multiplication $(f σ) (g τ) = f σ (g) σ τ$ .

[F1]

$H_{d}^{\deg}$ is presented by the relations of the Statement (Affine type-A presentations and charged cyclotomic quotients).

[F2]

$S_{d} = ⟨ s_{1}, \dots, s_{d - 1} ⟩$ with the Coxeter relations, and every word in the $s_{r}$ reduces to a standard word $w \in S_{d}$ ; a proper subword of a reduced expression has strictly smaller length (The symmetric group has the Coxeter presentation).

Proof

technique · direct

Given: The presentations of [F1], the field $F$ , the crossed product

$K ⋊ S_{d}$ , and the operator model of step 1.1.

1.1givenconstruct

The operator model. Let $S_{r}$ be the automorphism of $K$ swapping $z_{r}, z_{r + 1}$ and fixing the other variables, and put $δ_{r} : = z_{r} - z_{r + 1}$ . Let $z_{j}$ act on $K$ by multiplication and let $A_{r} : = S_{r} + \frac{S_{r} - 1}{δ_{r}} = (1 + \frac{1}{δ_{r}}) S_{r} - \frac{1}{δ_{r}} .$ For $p \in P$ the difference $S_{r} p - p$ is divisible by $δ_{r}$ in $P$ (write $p$ in the variables $z_{r}, z_{r + 1}$ with the remaining variables as scalars and use that $z_{r}^{a} z_{r + 1}^{b} - z_{r + 1}^{a} z_{r}^{b}$ is divisible by $δ_{r}$ ), so $A_{r}$ preserves $P$ ; hence the operators $z_{j}$ (multiplication) and $A_{r}$ generate a subalgebra of ${End}_{F} (K)$ and, restricted to $P$ , of ${End}_{F} (P)$ .

2.1F1algebra

Verification of the defining relations in the model. Direct substitution gives the mixed relations $A_{r} z_{r} - z_{r + 1} A_{r} = - 1$ , $A_{r} z_{r + 1} - z_{r} A_{r} = 1$ , and $A_{r} z_{j} = z_{j} A_{r}$ for $j \notin {r, r + 1}$ . Writing $a (t) : = 1 + 1 / t$ and $b (t) : = - 1 / t$ , so that $A_{r} = a (δ_{r}) S_{r} + b (δ_{r})$ , and using $S_{r} (δ_{r}) = - δ_{r}$ , the identity coefficient of $A_{r}^{2}$ is $a (δ_{r}) a (- δ_{r}) + b (δ_{r})^{2} = 1$ and its $S_{r}$ -coefficient is $a (δ_{r}) b (- δ_{r}) + b (δ_{r}) a (δ_{r}) = 0$ , so $A_{r}^{2} = 1$ ; distant $A_{r}$ commute because their permutations and coefficients involve disjoint variable pairs. For the braid relation put $u : = z_{r} - z_{r + 1}$ , $v : = z_{r + 1} - z_{r + 2}$ , $w : = u + v = z_{r} - z_{r + 2}$ , $S : = S_{r}$ , $T : = S_{r + 1}$ ; expanding $A_{r} A_{r + 1} A_{r}$ and $A_{r + 1} A_{r} A_{r + 1}$ in $K ⋊ S_{d}$ gives the six permutation coefficients $- \frac{u v + 1}{u v w} (1), \frac{a (u)}{v w} (S), \frac{a (v)}{u w} (T), - \frac{a (u) a (w)}{v} (S T), - \frac{a (v) a (w)}{u} (T S), a (u) a (w) a (v) (S T S),$ and the right expansion has the same six coefficients with $u, v$ and $S, T$ interchanged, so the two expansions agree. These are rational identities valid in every characteristic, including $2$ . Hence the assignment $z_{j} \mapsto$ multiplication by $z_{j}$ , $s_{r} \mapsto A_{r}$ defines an algebra homomorphism $H_{d}^{\deg} \to {End}_{F} (K)$ (and into ${End}_{F} (P)$ by step 1.1).

3.1F1F2step 2.1algebra

Spanning. In the model, $s_{r} p = (S_{r} p) s_{r} - (p - S_{r} p) δ_{r}^{- 1}$ for every polynomial $p$ , so every word in the generators rewrites as a linear combination of the ordered monomials $z^{a} w$ with $a_{j} \geq 0$ and $w \in S_{d}$ , the power-polynomial part being collected by commutativity and the $s$ -word reduced by [F2]. The same straightening performed in $H_{d}^{\deg}$ (available there because the model satisfies the defining relations, or by the same elementary computation in the presentation) shows that the ordered monomials span the algebra.

3.2F2step 2.1algebra

Independence. In the crossed-product expansion of a reduced product $A_{w}$ , the coefficient at $w$ is $c_{w} = \prod_{k = 1}^{m} (1 + 1 / (s_{i_{1}} \dots s_{i_{k - 1}}) (δ_{i_{k}})) \neq 0$ ; every other permutation has length less than $ℓ (w)$ , because it is represented by a proper subword. The distinct field automorphisms are independent over $K$ : from a shortest nonzero operator relation $\sum_{j} f_{j} σ_{j} = 0$ , compose with multiplication by a variable on which some $σ_{j}$ differs from $σ_{1}$ , and subtract the original relation multiplied by $σ_{1}$ of that variable. This gives a shorter nonzero relation; a one-term relation cannot annihilate $1$ . Thus the crossed product acts faithfully on $K$ . Now an $F$ -linear relation among the ordered monomials maps to $\sum_{w} f_{w} (z) A_{w} = 0$ , with $f_{w} (z) = \sum_{a} c_{a, w} z^{a} \in P$ . If some $f_{w}$ is nonzero, choose such a $w$ of maximal length. Its coefficient is $f_{w} c_{w} \neq 0$ and receives no contribution from any other maximal-length label or shorter label, a contradiction. Therefore every $f_{w} = 0$ , and polynomial coefficient uniqueness gives every $c_{a, w} = 0$ . This proves $F$ -linear independence of the ordered monomials, without asserting their impossible $K$ -linear independence.

4.1step 3.1step 3.2∎

Conclusion. Steps 3.1 and 2.1 show that the family ${x_{1}^{a_{1}} \dots x_{d}^{a_{d}} w : a_{j} \geq 0, w \in S_{d}}$ both spans and is linearly independent in $H_{d}^{\deg}$ over $F$ ; renaming the commuting generators $z_{j} \mapsto x_{j}$ and using the presentation [F1] gives the asserted $F$ -basis. No cyclotomic quotient, BK isomorphism or dimension theorem is used, and for $d = 0$ the family consists of the empty monomial alone.

Remarks

The model $A_{r} = S_{r} + (S_{r} - 1) / δ_{r}$ is the standard rational-operator model of the degenerate affine Hecke algebra; the only inversions are of the nonzero differences $δ_{r}$ , and the construction is valid in every characteristic. The independence proof uses finitely many automorphisms of a finitely generated field, so no choice principle is needed.

Depends on

Used by

Affine and cyclotomic type-A Hecke algebras Definition

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v4 (standard reference, not scraped)