Virtual Weyl and Specht symbols attached to beta sequences

Fix a beta length $L\ge 1$. For a partition $\nu$ with at most $L$ columns (that is, ${\nu }_1\le L$), put ${\beta }_i(\nu )={\nu }_i^{\mathrm{\prime }}+L-i(1\le i\le L)\mathrm{.}$ Its beta numbers are strictly decreasing nonnegative integers. Conversely, from any strictly decreasing nonnegative sequence $b$ recover the unique partition with conjugate parts ${\nu }_i^{\mathrm{\prime }}=b_i-L+i$, of degree $\mathrm{\mid }\nu \mathrm{\mid }={\sum }_i{b}_i-L(L-1)\mathrm{/}2$ (Partitions, English diagrams, and conjugation).

For arbitrary $b\in {\mathbb{Z}}^L$, first set all its alternating symbols to zero if it has repeated entries or a negative entry. Otherwise there is a unique sorting permutation $w$ taking it to decreasing order; let $\nu$ be the recovered partition and let $ϵ(b)=(-1{)}^{\mathrm{\ell }(w)}$. Uniqueness of the sorting permutation is claimed only in this distinct-entry case. In the fixed degree-$r$ Grothendieck groups define $[W^b{]}_{d,r}:=\{\begin{array}{ll}ϵ(b)[W^{\nu }],& \mathrm{\mid }\nu \mathrm{\mid }=r\text{ and }\mathrm{\ell }(\nu )\le d,\\ 0,& \text{otherwise},\end{array}[S^b{]}_r:=\{\begin{array}{ll}ϵ(b)[S_{\mathrm{M}}^{\nu }],& \mathrm{\mid }\nu \mathrm{\mid }=r,\\ 0,& \text{otherwise}\mathrm{.}\end{array}$ The notation $[{\mathrm{\Delta }}^b]$ is synonymous with $[W^b{]}_{d,r}$ when $d,r$ are fixed. The ordinary Dipper-James translation conjugates every partition label and pulls every compared module back by the same $\theta$ (The type-A Hecke Specht module and its cell form, The type-A q-Schur algebra and its Weyl modules).

For a composition $\mu$ define the signed semistandard number directly by $d_{\mu }(b):=\{\begin{array}{ll}ϵ(b)\mathrm{\mid }{T}_0(\nu ,\mu )\mathrm{\mid },& b\text{ has distinct nonnegative entries and }\mathrm{\mid }\nu \mathrm{\mid }=\mathrm{\mid }\mu \mathrm{\mid },\\ 0,& \text{otherwise}\mathrm{.}\end{array}$ Here the tableau alphabet is the length of $\mu$; a shape with too many rows automatically has no such tableau (Semistandard tableaux and Kostka numbers). This function is alternating in the beta entries. It also obeys the branching identity $d_{\mu }(b)={\sum }_{\begin{array}{c}C\subseteq \{1,\dots ,L\}\\ \mathrm{\mid }C\mathrm{\mid }=z\end{array}}{d}_{\bar{\mu }}(b-{\mathbf{1}}_C),$ where $z$ is the last nonzero part of $\mu$ and $\bar{\mu }$ deletes that part. The selected entries are decreased by one, not replaced by $-1$. For a decreasing beta sequence this is the deletion of the largest semistandard entries: they occupy a horizontal strip, at most one box in each selected column. Sorting extends the identity to distinct arbitrary sequences; if entries repeat, terms selecting just one of the equal entries cancel in pairs, while terms selecting both or neither vanish. Negative entries remain negative after decrementing. Thus the identity also holds in all zero-symbol cases. At $\mu =0$ its value is the sorting sign for a permutation of $(L-1,\dots ,0)$ and is zero otherwise, so the recursion terminates without another unproved extension rule.

In the formulas of this page take $L=r$ when $r>0$. A nonzero degree-$r$ partition has width at most $r$, so this length suffices; the Weyl cutoff $\mathrm{\ell }(\nu )\le d$ is retained even when $d<r$. A balanced displacement $+h{\mathbf{e}}_b-h{\mathbf{e}}_c$ preserves total degree. An unbalanced change is not silently treated as a class in the same fixed-degree group. The empty degree is handled by the empty beta sequence and unit symbol. No choice principle is used.