The graded Specht module by homogeneous standard tableaux

The ideal and the cyclic generator. The span $K(\mu )$ of the Murphy elements indexed by standard pairs of shape strictly dominating $\mu$ is a two-sided ideal by the cellularity theorem thm-arbitrary-level-cyclotomic-hecke-murphy-basis-is-cellular, and the class $z_{\mu }=m_{t^{\mu }{t}^{\mu }}+K(\mu )$ is the canonical generator of the cell module [F2]. The left action is obtained from the right cellular datum by $h\cdot m=m{h}^{\ast }$, so $S(\mu )=H_d^{\mathrm{\Lambda }}{z}_{\mu }$ is a left module; no right module is identified with a left module without this conversion [F2].

The initial weight vector and block. The row-filled tableau $t^{\mu }$ dominates every standard tableau of its shape at every prefix. Thus the triangular JM formula of The Hecke Jucys-Murphy action is dominance triangular has no off-diagonal terms at this vector. Star fixes $L_r$, so conversion to the left cell module gives $L_r{z}_{\mu }=[c-r{]}_{\xi }{z}_{\mu }$ for the node $(r,c)$ containing that entry. The multiplicative JM recurrence $X_{a+1}={\xi }^{-1}{T}_a{X}_a{T}_a$ and the quadratic relation give $X_a=1+(\xi -1)L_a$: its inductive step is $L_{a+1}={\xi }^{-1}{T}_a{L}_a{T}_a+{\xi }^{-1}{T}_a$. Hence $X_a{z}_{\mu }={\xi }^{i_a}{z}_{\mu }$ by $1+(\xi -1)[m{]}_{\xi }={\xi }^m$, valid also for negative content. The weight idempotents and $y_a=(1-{\xi }^{-i_a}{X}_a)e(i)$ now give $e(i_{t^{\mu }})z_{\mu }=z_{\mu }$ and $y_a{z}_{\mu }=0$. Since the vectors generated from $z_{\mu }$ have sequences obtained by permuting its residues, the whole cyclic module lies in content block $\alpha =\mathrm{cont}(\mu )$.

The vectors. For each standard $\mu$-tableau $T$ fix a reduced decomposition $w_T=s_{r_1}\cdots s_{r_m}$; the elements $z_T=T_{w_T}{z}_{\mu }$ and $v_T={\psi }_{w_T}{z}_{\mu }$ are defined by left multiplication in the Hecke and KLR presentations, and $T_{w_T}$ is independent of its Hecke reduced expression, whereas ${\psi }_{w_T}{z}_{\mu }$ can depend on the chosen reduced word: the KLR braid relation has correction terms [F5]. The resulting family is defined for the selected reduced words, and the discharge theorem proves that every selected family gives the same graded subspaces; it does not assert that the individual vectors are word-independent [F5].

The degree recursion is well defined. For a standard tableau $T$ of size $d>0$, $T_{\le d-1}$ is a standard tableau of the smaller shape and the node $A$ of $T$ containing $d$ is removable in that shape, so the recursion $\mathrm{deg}(T)=\mathrm{deg}(T_{\le d-1})+d_A(\mu )$ is a finite induction on $d$ with values in $\mathbb{Z}$ depending only on $T$ and $\mu$; the declared boundary count is evaluated in the same residue convention at every stage [F1, F3].

The ungraded comparisons and Choice. Forgetting the grading, the star-contragredient duality of thm-dj-bridge-murphy-cell-is-the-same-shape-specht-dual identifies the left cell module with the same-shape Specht dual, and the signed-conjugate comparison of thm-dj-bridge-ordinary-specht-is-the-signed-conjugate-murphy-cell gives $S(\mu )\cong P({\mu }^{\mathrm{\prime }}{)}^{\theta }$ after conversion; since $\theta$ commutes with star by [F4], both comparisons keep their stated positions and neither asserts untwisted conjugate identification [F4]. Every construction selects finite sets of reduced decompositions and finite basis data, so no choice principle is used.