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The higher-level wedge-tensor Fock module

Definition

Work with the type- $A$ coordinate lattice and the dictionary of The Fock-space convention dictionary for finite type-A quantum modules; weights are written in the coordinates $δ_{a} = ε_{a}$ .

Fix a set $I$ of consecutive residues (for the finite box model $I = {k_{0}, \dots, k_{0} + N - 2}$ and $I^{+} = I \cup (I + 1) = {k_{0}, \dots, k_{0} + N - 1}$ ; $I$ may be unbounded below or above), a positive level $l$ , and $Λ = Λ_{m_{1}} + \dots + Λ_{m_{l}} \in P^{+}, m_{1} \geq m_{2} \geq \dots \geq m_{l},$ where $m_{j} \in I^{+}$ and, if $I$ has lower endpoint $k$ , $Λ_{m} = \sum_{k \leq a \leq m} δ_{a}$ . When $I$ is unbounded below, $Λ_{m}$ is the ground semi-infinite fundamental weight with the finite-coordinate-difference interpretation below. For $I$ bounded below put $n_{j} = m_{j} + 1 - \inf (I)$ boxes in column $j$ and $n = \sum_{j} n_{j}$ ; for $I$ unbounded below use semi-infinite columns with a common finite tail below, as explained after the weight definition.

The higher-level wedge-tensor Fock module is $F (Λ) : = V (Λ_{m_{1}}) \otimes V (Λ_{m_{2}}) \otimes \dots \otimes V (Λ_{m_{l}}),$ the tensor product of the fundamental modules of the dictionary item. When $I$ is bounded below, $V (Λ_{m_{j}})$ is the fundamental integrable highest weight module of highest weight $Λ_{m_{j}}$ in the library algebra. When $I$ is unbounded below, $V (Λ_{m_{j}})$ is the semi-infinite wedge $Λ^{m_{j} + \infty} V$ with $V$ the natural module for the locally finite type- $A$ algebra on $I$ ; its basis is the set of normally ordered wedges $v_{i_{1}} \land v_{i_{2}} \land \dots$ with $i_{1} > i_{2} > \dots$ and the tail $i_{k} = m_{j} + 1 - k$ for all sufficiently large $k$ .

A $Λ$ -tableau $A$ fills the diagram of $Λ$ — the columns $1, \dots, l$ of heights $n_{j}$ , with rows indexed by $I^{+}$ — with entries in $I^{+}$ ; it is column-strict if in each column the entries strictly increase from bottom to top and all but finitely many entries of row $i$ equal $i$ , and standard if in addition the entries weakly increase from left to right in each row. Write ${C o l}^{Λ}$ , respectively ${S t d}^{Λ}$ , for the sets of column-strict, respectively standard $Λ$ -tableaux, and $A_{Λ}$ for the ground-state tableau with row $i$ filled entirely with $i$ .

For $A \in {C o l}^{Λ}$ the column monomial is $M_{A} : = (v_{A (m_{1}, 1)} \land v_{A (m_{1} - 1, 1)} \land \dots) \otimes \dots \otimes (v_{A (m_{l}, l)} \land v_{A (m_{l} - 1, l)} \land \dots),$ with each wedge normally ordered (the entries strictly decrease from top to bottom inside a column), and ${w t}_{j} (A) = \sum_{m_{j} \geq i \in I^{+}} δ_{A (i, j)}, w t (A) = {w t}_{1} (A) + \dots + {w t}_{l} (A), {C o l}_{α}^{Λ} = {A : w t (A) = Λ - α}$ for $α \in Q^{+}$ . When a column is semi-infinite, the displayed sum means its ground fundamental weight $Λ_{m_{j}}$ plus the finite sum of the coordinate differences $δ_{A (i, j)} - δ_{i}$ over the finitely many positions where $A$ differs from the ground state, evaluated on each finite-support coroot or torus input; no divergent infinite coordinate sum is asserted.

The Bruhat order on ${C o l}^{Λ}$ is defined by $A \leq B$ iff $w t (A) = w t (B)$ and ${w t}_{1} (A) + \dots + {w t}_{j} (A) \geq {w t}_{1} (B) + \dots + {w t}_{j} (B)$ in root dominance for every $j < l$ : the difference of the two prefix weights is a nonnegative integral combination of the simple roots $α_{a} = δ_{a} - δ_{a + 1}$ . This is the coordinate-prefix criterion verified below, and is defined for these weight vectors rather than only for partitions (Dominance order on partitions); then $A < B$ whenever $B$ is obtained from $A$ by swapping entries $a < b$ in columns $i < j$ . The column reading of a tableau (list the columns from left to right, each column read top to bottom) gives the passage to partitions and to the Young-diagram conventions of Partitions, English diagrams, and conjugation and Tableaux and standard tableaux; the removable- and addable-node notation of Removable and addable nodes is used for the box moves of a tableau.

The vector $v_{Λ} : = M_{A_{Λ}}$ is called the vacuum vector. Weight bookkeeping, the uniqueness of the monomial of weight $Λ$ and the bijectivity of the column reading are verified below; the column-monomial theorem records the full integral tensor-basis statement. The well-definedness verification here constructs each fundamental factor explicitly without treating a citation to integrability as a wedge-basis theorem.

Facts & Assumptions

Given: The coordinate lattice and dictionary of The Fock-space convention dictionary for finite type-A quantum modules, a level $l$ and a dominant weight $Λ = Λ_{m_{1}} + \dots + Λ_{m_{l}}$ with $m_{1} \geq \dots \geq m_{l}$ ; the sets ${C o l}^{Λ}, {S t d}^{Λ}$ with the ground tableau $A_{Λ}$ , the monomials $M_{A}$ , the weights ${w t}_{j}$ and $w t$ , and the Bruhat order of the Definition.

[F1]

Dictionary and coordinate weights (The Fock-space convention dictionary for finite type-A quantum modules): the weight lattice is $⨁_{a \in I^{+}} Z δ_{a}$ with the full central coordinate, the natural highest weight of the fundamental factor is $δ_{k}$ , the dual highest weight is $- δ_{r}$ , and the fundamental wedge carries $δ_{k} + \dots + δ_{m}$ .

[F2]

Partitions and dominance (Partitions, English diagrams, and conjugation, Dominance order on partitions, Conjugation reverses dominance): dominance $⊵$ on partitions of a fixed integer is the termwise dominance of prefix sums, and conjugation reverses it.

[F3]

Tableau conventions (Tableaux and standard tableaux, Semistandard tableaux and Kostka numbers, Removable and addable nodes): standard and semistandard tableaux, shapes and removable/addable nodes in the Young-diagram normalization.

[F4]

The dictionary supplies the natural representation and its Hopf action, and the existence and uniqueness of its integrable highest-weight quotients. The elementary ordered-wedge construction is proved in step 1.1, not imported from the definition of integrability.

Verification

1.1F1F4constructalgebra

Elementary fundamental modules. On a finite coordinate interval $[k, r]$ let $W_{n}$ have vectors $w_{S}$ indexed by $n$ -element subsets $S$ . Put $D_{a} w_{S} = s^{1_{a \in S}} w_{S}$ , and let $E_{a}$ replace $a + 1 \in S$ by $a \notin S$ , while $F_{a}$ makes the opposite replacement, both with coefficient one and zero otherwise. The toral relations follow from the changed coordinate; $[E_{a}, F_{a}]$ is $1$ , $- 1$ or zero on the three possible $a, a + 1$ occupation patterns, exactly the toral fraction. Different-root mixed commutators vanish by the same replacement test. Distant replacements commute; for adjacent roots $E_{a}^{2} = F_{a}^{2} = 0$ and $E_{a} E_{b} E_{a} = F_{a} F_{b} F_{a} = 0$ , since the last replacement encounters the already occupied endpoint. These are the adjacent Serre relations. Distinct subset weights are separated by the toral operators, and successive raising replacements take any subset to ${k, \dots, k + n - 1}$ ; lowering generates every subset from it. Thus this module is integrable and irreducible, of highest weight $Λ_{k + n - 1}$ , and is the fundamental module in the dictionary. The ordered wedge symbol denotes this model; the primitive antisymmetrizer realizes it integrally, without any assertion that the raw tensor lower chart makes that embedding based.

2.1F2F3step 1.1

Root rows and physical rows. In column $j$ the root row runs from $k$ to $m_{j}$ , whereas its physical height is $n_{j} = m_{j} + 1 - k$ . The monomial therefore starts at $A (m_{j}, j)$ and descends through $A (m_{j} - 1, j), \dots, A (k, j)$ . Its entries are strictly decreasing in that order and define a nonzero $w_{S}$ from step 1.1. Physical row $h$ means root row $k + h - 1$ ; its row length is $# {j : n_{j} \geq h}$ . Hence the ordinary partition diagram is the conjugate of $(n_{1}, \dots, n_{l})$ , not of the possibly negative multicharge. Column reading is a bijection to tuples of strictly decreasing entry blocks of these fixed lengths; the standard row inequalities are the corresponding semistandard inequalities.

3.1F1step 1.1step 2.1

Weights and the vacuum. The toral formula in step 1.1 gives precisely ${w t}_{j} (A)$ and their sum on the tensor monomial. The ground column is ${k, \dots, m_{j}}$ , with weight $Λ_{m_{j}}$ . Every other column weight differs from its ground by a nonzero element of the positive root cone: repeatedly raise its subset to the initial segment, decreasing its weight difference by a simple root at every step. The tensor top weight can therefore occur only when every factor is its ground vector. Its weight space is one-dimensional, spanned by $v_{Λ}$ . This argument uses the explicit finite factor construction, not a yet unproved global-basis inclusion.

3.2F1F2step 2.1algebra

The prefix order. Use root dominance: a weight difference is nonnegative when it is a nonnegative integral combination of $α_{a} = δ_{a} - δ_{a + 1}$ ; on fixed-total-coordinate weights this agrees with the usual coordinate prefix criterion. Reflexivity and transitivity are immediate; equality of every factor-prefix weight forces equality of each column subset, hence of the column-strict tableaux, proving antisymmetry. If $A$ has $a$ in an earlier column $i$ and $b > a$ in a later column $j$ , and $B$ swaps them, the prefix difference ${w t}_{\leq u} (A) - {w t}_{\leq u} (B)$ is $δ_{a} - δ_{b} = \sum_{v = a}^{b - 1} α_{v}$ for $i \leq u < j$ and zero otherwise. Thus $A < B$ .

4.1F1F4step 1.1step 2.1step 3.1step 3.2

Infinite intervals. Every fixed tableau differs from its ground only finitely many times. Choose an actual finite lower cut below all those changes, never an undefined $\inf I$ when $I$ is unbounded below, and a finite upper cut above all displayed entries and charges. Appending a common ground tail gives the same action of every root generator in the common subinterval: its adjacent occupation pattern is unchanged. The finite weight differences and all prefix comparisons are unchanged. The compatible union constructs the semi-infinite factors and tensor module; each fixed finite-support calculation is one of the finite computations above. Highest-weight uniqueness and the ground argument remain valid on those finite cuts.

5.1step 1.1step 2.1step 3.1step 3.2step 4.1∎

Conclusion. The module, tableau labels, column monomials, root-coordinate weights and prefix order are well defined, and the vacuum is the unique normalized highest-weight tensor monomial. Ordinary physical partitions use the shifted column heights. All admitted consecutive intervals and multicharges are retained; no raw-chart basedness or additional choice is asserted.

Depends on

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Sources

Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)