{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:00Z","timestamp":1753893840613,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\\mathbf i \\in [r]^n$ be a word and let $\\boldsymbol{\\ell} = (\\ell_1,\\dots,\\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\\mathbf i$ and $\\boldsymbol{\\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\\mathbf i$ and $\\boldsymbol{\\ell}$. In recent work, the author and Harada described precisely when the Grossberg\u2013Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\\boldsymbol{\\ell}$ comes from a weight $\\lambda$ of $G$. However, not every integer sequence $\\boldsymbol{\\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg\u2013Karshon twisted cubes associated with any word $\\mathbf i$ and any integer sequence $\\boldsymbol{\\ell}$ using the combinatorics of $\\mathbf i$ and $\\boldsymbol{\\ell}$. Indeed, we prove that the Grossberg\u2013Karshon twisted cube is untwisted precisely when $\\mathbf i$ is hesitant-jumping-$\\boldsymbol{\\ell}$-walk-avoiding.<\/jats:p>","DOI":"10.37236\/9278","type":"journal-article","created":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:36Z","timestamp":1599187656000},"source":"Crossref","is-referenced-by-count":1,"title":["Grossberg\u2013Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance"],"prefix":"10.37236","volume":"27","author":[{"given":"Eunjeong","family":"Lee","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,8,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p34\/8152","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p34\/8152","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:37Z","timestamp":1599187657000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,21]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9278","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,8,21]]},"article-number":"P3.34"}}