The regular integral dot-orbit labelling convention

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ of type $A_{n-1}$ with Cartan $\mathfrak{h}$, positive system ${\mathrm{\Phi }}^{+}$, Weyl vector $\rho$, Weyl group $W$ (identified with $S_n$) and dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$. Fix a dominant regular integral weight $\lambda$, i.e. $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{\ne }0$ for every root and $\lambda$ integral, and let ${\mathcal{O}}_{\lambda }$ be the corresponding regular integral block in the conventions of The regular integral block and its antidominant weight: standard objects $\mathrm{\Delta }(w\cdot \lambda )=M(w\cdot \lambda )$, simples $L(w\cdot \lambda )$, indecomposable projectives $P(w\cdot \lambda )$ for $w\in W$, with $\lambda =e\cdot \lambda$ the unique dominant and $w_0\cdot \lambda$ the unique antidominant element and $\mathrm{\Delta }(w_0\cdot \lambda )=L(w_0\cdot \lambda )$.

This item freezes the following conventions, each inherited from the cited suppliers and used verbatim below.

(i) Multiplicity conventions. $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))$ is the Verma-flag multiplicity of Finite Verma flags and their multiplicities and $[M(y\cdot \lambda ):L(w\cdot \lambda )]$ the composition multiplicity, and $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))=[M(y\cdot \lambda ):L(w\cdot \lambda )]$ by BGG reciprocity (BGG reciprocity).

(ii) Bruhat reversal convention. Along the bijection $W\ni w\mapsto w\cdot \lambda$ the Bruhat order runs opposite to the dominance order, in the two exact senses used on this page: $y\le w$ implies $w\cdot \lambda \le y\cdot \lambda$ in the dominance order, and every nonzero multiplicity forces the index direction, i.e. $[M(y\cdot \lambda ):L(w\cdot \lambda )]\mathrm{\ne }0$ implies $y\le w$ in the Bruhat order. The first sense is the inclusion direction of Dominant integral dot translates embed canonically in the Verma module (for $w\ge y$ one has $M(w\cdot \lambda )\subseteq M(y\cdot \lambda )$, so the highest weight of the submodule is the lower one), and the second is Jordan-Holder factors of Verma modules dominate the head (BGG 8.12), which confines every composition factor of $M(y\cdot \lambda )$ to labels $w\ge y$. The full statement that the two orders are isomorphic as posets is not needed here and is not asserted; only these two directions, together with the finiteness of intervals of the Bruhat order (Basic properties of the Bruhat order on $S_n$), are used.

(iii) Hecke and Kazhdan–Lusztig normalization. The Kazhdan–Lusztig basis ${\underset{‾}{H}}_w={\sum }_{y\le w}{p}_{y,w}{H}_y$ of the normalized type-$A$ Hecke algebra (Existence and uniqueness of the Kazhdan–Lusztig basis), with $p_{y,w}=v^{\mathrm{\ell }(w)-\mathrm{\ell }(y)}{P}_{y,w}(v^{-2})$ in the classical $q=v^{-2}$ normalization (Kazhdan–Lusztig polynomials in the classical $q$-normalization), support $p_{y,w}=0$ unless $y\le w$, diagonal $p_{w,w}=1$, and simultaneous inversion symmetry $p_{y^{-1},w^{-1}}=p_{y,w}$.

(iv) Soergel label and character dictionary. Write $R_x$ for the polynomial standard graph bimodule supported at $x$, and $B_x$ for the normalized indecomposable polynomial Soergel bimodule, which is generally not a standard graph bimodule. With right reduction $\bar{B}=B{\otimes }_S\mathbb{C}$ and left coinvariant-module values of $V$, the projective dictionary is $V(P(w\cdot \lambda ))\cong {\bar{B}}_{w^{-1}}$ (Indecomposable projectives correspond to indecomposable Soergel modules). The graph label $x$ corresponds to the standard $\mathrm{\Delta }(x^{-1}\cdot \lambda )$. The individual support ranks are measured on the polynomial lift before reduction, as in Verma-flag multiplicities are standard-filtration ranks under V; no separated graph-support filtration of the reduced module is asserted.

For degree bookkeeping use precisely the standard-generator normalization of Standard graph bimodules, support filtrations and characters: ${\mathrm{\Delta }}_x(d)=R_x(d-\mathrm{\ell }(x))$, and $r_{x,d}(B)$ counts these normalized polynomial standard layers. Thus the coefficient polynomial is $m_x(B;v):={\sum }_d{r}_{x,d}(B)v^d,h_{\mathrm{\Delta }}(B)={\sum }_x{m}_x(B;v){\tilde{T}}_x,{\tilde{T}}_x=v^{\mathrm{\ell }(x)}{T}_x\mathrm{.}$ Here ${\tilde{T}}_x$ is the normalized standard Hecke element $H_x$ used in clause (iii) and The normalized type-A Hecke algebra and its bar involution. The published Soergel split-Grothendieck-ring convention instead names its simple Kazhdan--Lusztig generators $H_i=v(T_i+1)={\tilde{T}}_{s_i}+v$; these must not be confused with the standard element $H_{s_i}=v{T}_i$ (The split Grothendieck group of the Soergel category is the type-A Hecke algebra). The standard-character supplier The type-A standard character is multiplicative identifies $h_{\mathrm{\Delta }}$ with that split-Grothendieck-ring isomorphism. This records the coefficient normalization, and does not infer the image of a general indecomposable $B_x$ by multiplying simple generators. In particular the character theorem must be used to identify $h_{\mathrm{\Delta }}(B_x)$ with ${\underset{‾}{H}}_x$; a product of simple Soergel classes is a Bott--Samelson class and need not be $[B_x]$. The inverse labels and all rank comparisons above use auxiliary polynomial gradings only; forgetting them gives the ordinary projective dictionary.