FLOW POLYTOPES OF PARTITIONS

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1 FLOW POLYTOPES OF PARTITIONS KAROLA MÉSZÁROS, CONNOR SIMPSON, AND ZOE WELLNER Abstract. Recent progress on flow polytopes ndcates many nterestng famles wth product formulas for ther volume. These product formulas are all proved usng analytc technques. Our work breaks from ths pattern. We defne a famly of closely related flow polytopes F (λ,a) for each partton shape λ and netflow vector a Z n >0. In each such famly, we prove that there s a polytope (the lmtng one n a sense) whch s a product of scaled smplces, explanng ther product volumes. We also show that the combnatoral type of all polytopes n a fxed famly F (λ,a) s the same. When λ s a starcase shape and a s the all ones vector the latter results specalzes to a theorem of the frst author wth Morales and Rhoades, whch shows that the combnatoral type of the Tesler polytope s a product of smplces. 1. Introducton The Catalan numbers, C n = 1 n+1( n 2), n Z 0, are well known for countng a plethora of combnatoral objects; see [6, Ex. 6.19] for hundreds of nterpretatons. Naturally then, f an nteger polytope has volume dvsble by a product of consecutve Catalan numbers, one would hope for a combnatoral explanaton of such a phenomenon. The latter sentment ran nto obstacles wth several flow polytopes, namely the (type A) Chan-Robbns-Yuen polytope [2], ts type C and D generalzatons [4], as well as the Tesler polytope [5]. Our work s nspred by the Tesler polytope (whch s the flow polytope of the complete graph wth netflow vector all ones) F Kn+1 (1), whch we explan below can be assocated wth a starcase partton. Two of the known man results known about F Kn+1 (1) are as follows. We defne F Kn+1 (1), and flow polytopes n general, n Secton 2. Theorem 1.1. [5, Theorem 1.9] The normalzed volume of the Tesler polytope F Kn+1 (1) equals ( n ) 2! 2 ( n 2) vol F Kn+1 (1) = n =1! (1.1) n 1 = SY T (n 1,n 2,...,1) C, where C s the th Catalan number and SY T (n 1,n 2,...,1) s the number of Standard Young Tableaux of starcase shape (n 1, n 2,..., 1). Theorem 1.2. [5, Corollares 2.8 & 2.9] The face poset of the Tesler polytope F Kn+1 (1) s somorphc to the face poset of the Cartesan product of smplces 1 =0 Mészáros s partally supported by a Natonal Scence Foundaton Grant (DMS ). 1

2 2 KAROLA MÉSZÁROS, CONNOR SIMPSON, AND ZOE WELLNER 2 n 1. In partcular, the h-polynomal of the Tesler polytope F Kn+1 (1) s the Mahonan dstrbuton ( n 2) h x = [n]! x = (1 + x)(1 + x + x 2 ) (1 + x + x x n 1 ). =0 For each partton λ and vector a we construct a famly of flow polytopes F (λ,a), whch we defne n Secton 2.2. The Tesler polytope F Kn+1 (1) belongs to F ((n 1,n 2,...,1),1). We prove the followng general theorems about the famles F (λ,a). The lmtng polytope F(λ,a) lm s defned n Secton 3. Theorem 3.3. Let λ be a partton, n λ 1 + l(λ), and a (Z >0 ) n a vector of postve ntegers. The lmtng polytope of F (λ,a) s ntegrally equvalent to a product of scaled smplces a 1 λ1 a l(λ) λl(λ). Consequently, t has normalze volume (1.2) vol F lm (λ,a) = [l(λ)] λ! [l(λ)] Theorem 4.7. Let λ = (λ 1,..., λ k ) be a partton, n an nteger such that n λ for all [l(λ)], and a Z n >0 a netflow vector. The face posets of the polytopes belongng to F (λ,a) are somorphc to the face poset of the Cartesan product of smplces λ1 λ2 λk. In partcular, the h-polynomal of the polytopes belongng to F (λ,a) s k =1 λ =0 h x = k [λ ] x = =1 k λ 1 In partcular, we see that Theorem 1.2 s a specal case of Theorem 4.7 for F Kn+1 (1) whch belongs to F ((n 1,n 2,...,1),1). Also notce the smlar volumes for F Kn+1 (1) (Theorem 1.1) and F((n 1,n 2,...,1),1) lm (Theorem 3.3); they are off by a 2) factor of 2(n n!. We spell ths curous fact out n the next corollary. =1 j=0 a λ λ! Corollary 1.3. vol F Kn+1 (1) = 2(n 2) n! vol F lm ((n 1,n 2,...,1),1) The outlne of ths paper s as follows. In Secton 2 we cover the necessary background and defne the class F (λ,a). In Secton 3 we defne the lmtng polytope and prove Theorem 3.3. Secton 4 s devoted to provng Theorem 4.7. F lm (λ,a) x j 2. Background and defntons 2.1. Flow polytopes and Kostant partton functons. The exposton of ths secton follows that of [4]; see [4] for more detals. Let G be a (loopless) graph on the vertex set [n + 1] wth N edges. To each edge (, j), < j, of G, assocate the postve type A n root v(, j) = e e j, where e s the th standard bass vector n R n+1. Let S G := {{v 1,..., v N }} be the multset of roots correspondng to the multset of edges of G. Let M G be the (n + 1) N matrx whose columns are the vectors n S G. Fx an nteger vector a = (a 1,..., a n+1 ) Z n+1 whch we call the netflow and for whch we requre that.

3 FLOW POLYTOPES OF PARTITIONS 3 a n+1 = n =1 a. An a-flow f G on G s a vector f G = (b k ) k [N], b k R 0 such that M G f G = a. That s, for all 1 n + 1, we have (2.1) e=(g<) E(G) b(e) + a = e=(<j) E(G) Defne the flow polytope F G (a) assocated to a graph G on the vertex set [n + 1] and the nteger vector a = (a 1,..., a n+1 ) as the set of all a-flows f G on G,.e., F G = {f G R N 0 M Gf G = a}. The flow polytope F G (a) then naturally lves n R N, where N s the number of edges of G. Note that n order for F G (a) to be nonempty, t must be that n+1 =1 a = 0. For ths reason, we also wrte F G (a 1,..., a n ) := F G (a 1,..., a n, n =1 a ). The vertces of the flow polytope F G (a) are the a-flows whose supports are acyclc subgraphs of G [3, Lemma 2.1]. Recall that the Kostant partton functon K G evaluated at the vector b Z n+1 s defned as (2.2) K G (b) = # {(c k ) k [N] k [N] b(e) c k v k = b and c k Z 0 }, where [N] = {1, 2,..., N}. The generatng seres of the Kostant partton functon s (2.3) K G (b)x b = (1 x x 1 j ) 1, b Z n+1 (,j) E(G) where x b = x b1 1 xb2 2 xbn+1. In partcular, (2.4) K Kn+1 (b) = [x b ] 1 <j n+1 (1 x x 1 j ) 1. Assume that a = (a 1, a 2,..., a n ) satsfes a 0 for = 1,..., n. Let a = (a 1, a 2,..., a n, n =1 a ). The generalzed Ldsk formulas of Baldon and Vergne state that for a graph G on the vertex set [n + 1] wth N edges we have Theorem 2.1. [1, Theorem 38] (2.5) vol F G (a ) = ( ) N n a 1 1 1, 2,..., an n K G ( 1 t G 1, 2 t G 2,..., n t G n ), n and (2.6) K G (a ) = ( a1 + t G )( 1 a2 + t G ) ( an + t G n n ) K G ( 1 t G 1, 2 t G 2,..., n t G n ), where both sums are over weak compostons = ( 1, 2,..., n ) of N n wth n parts whch we denote as = N n, l() = n. The graph G s the restrcton of G to the vertex set [n]. The notaton t G, [n], stands for the outdegree of vertex n G mnus 1. The notaton vol stands for normalzed volume. Recall that the Ehrhart polynomal (P, t) of an nteger polytope P R m counts the number of nteger ponts of dlatons of the polytope, (P, t) := #(tp Z m ). Its leadng coeffcent s the volume of the polytope. The normalzed volume vol(p ) of a d-dmensonal

4 4 KAROLA MÉSZÁROS, CONNOR SIMPSON, AND ZOE WELLNER polytope P R m s the volume form whch assgns a volume of one to the smallest d-dmensonal nteger smplex n the affne span of P. In other words, the normalzed volume of a d-dmensonal polytope P s d! tmes ts volume The famly F (λ,a). We start by defnng a famly of graphs assocated to the partton λ. Gven a partton λ, let Y be the left-justfed Young dagram correspondng to λ. Pck an nteger n such that n λ for all [l(λ)]. We can place Y nsde the upper trangle (not ncludng the dagonal) of an n n matrx M, wth the top and rght edges of Y flush wth the top and rght edges of M. Now, let Y be the set of entres (, j) of M that le nsde Y, and defne G(λ, n) to be the drected graph ( G(λ, n) := [n + 1], {(, n + 1) : [n]} Y ). Example 2.2. Constructon of G((2, 1, 1), 5)) Fgure 1. From left to rght: the left-justfed Young dagram of λ = (2, 1, 1), the dagram n a 5 5 matrx, and the correspondng graph on sx vertces. For a vector a Z m >0 wth m λ 1 + l(λ), defne the famly F (λ,a) := { F G(λ,n) (a) : max(λ 1, l(λ)) < n Z }. Note that there s a small abuse of notaton n the defnton above: f n m, then a wll have too many or too few entres to serve as a netflow for many G(λ, n). When n m, then we can just use the frst n entres of a. For n > m, we show n Secton 3 that the choce of addtonal entres s rrelevant: any element of Z n >0 whose frst m entres match those of a wll product essentally the same polytope. More precsely, we prove that all the above mentoned polytopes are ntegrally equvalent. Recall that nteger polytopes P R m and Q R k are ntegrally equvalent f there s an affne transformaton f : R m R k such that f maps P bjectvely onto Q and f maps Z m aff(p) bjectvely onto Z k aff(q), where aff denotes affne span. If two polytopes are ntegrally equvalent, then they have the same combnatoral type as well as the same volume and more generally the same Ehrhart polynomal. Observe that for any n Z >0 and λ = (n 1, n 2,..., 1), G(λ, n) = K n+1. Settng a = 1, t follows that the Tesler polytope F Kn+1 (1) belongs to F ((n 1,n 2,...,1),1). 3. The lmtng polytopes of F (λ,a) In ths secton we defne the lmtng polytope of the famly F (λ,a) for any partton λ and netflow vector a. We then establsh the combnatoral structure and the volume of these lmtng polytopes.

5 FLOW POLYTOPES OF PARTITIONS 5 One can easly see the need to defne a lmtng polytope of F (λ,a) from the followng data on the normalzed volumes of the members of the famly F ((4,3,2,1),1) : n vol F G((4,3,2,1),n) (1) One mmedately notces that the volume of the polytopes n queston appears to stablze for large n. Ths s not a concdence, and s n fact a general feature of polytopes n F (λ,a), as we show n ths secton. For a partton λ and a Z n >0, defne the lmtng polytope of the famly F (λ,a), denoted F(λ,a) lm, to be the polytope F G(λ,l(λ)+λ 1)(a). We prove n Lemma 3.1 that for all n l(λ) + λ 1 we have that F G(λ,n) (a) and F G(λ,l(λ)+λ1)(a) are ntegrally equvalent; thus any one of F G(λ,n) (a) wth n l(λ) + λ 1 can be thought of as F(λ,a) lm Structure and Volume of the Lmtng Polytope. Gven a graph G = G(λ, n), for each vertex [n], let G = ([n + 1], {(, j) E(G) : < j} {(j, n + 1) : j [n]}) be the subgraph of G graph obtaned by restrctng E(G) to those edges that come out of vertex or go to the snk. Lemma 3.1. Let λ be a partton, let n l(λ) + λ 1, let G and G be as above for [n], and let a Z n >0. Then F G(λ,n) (a) s ntegrally equvalent to n =1 F G (a): n F G(λ,n) (a) F G (a). =1 Proof. Defne the map ϕ : F G (a) n =1 F G (a) by ϕ(f) = (f 1,..., f n ) where f : E(G ) R s defned by f(, j), (p, q) = (, j) f (p, q) = a p + f(, p), q = n + 1 and (, p) E(G ) a p, q = n + 1 and (, p) E(G ) The nverse of map ϕ s ϕ 1 : n =1 F G F G defned by ϕ 1 (f 1,..., f n ) = f where f(p, q) = f p (p, q), thus ϕ s a bjecton between F G(λ,n) (a) and n =1 F G (a). Moreover, ϕ can be extended to an affne map mappng the nteger ponts of the affne span of F G(λ,n) (a) bjectvely to the nteger ponts of the affne span of n =1 F G (a), concludng the proof. We now show that the polytopes F G (a) appearng n Lemma 3.1 are very specal: Lemma 3.2. For [l(λ)], a Z n >0, F G (a) s ntegrally equvalent to a λ, a scaled smplex of dmenson λ. For l(λ) < < n + 1, F G (a) s a pont. Proof. Let [l(λ)]. Defne ϕ : F G (a) a λ by ϕ (f ) = v R λ+1, where v j = f (, n + 2 j). To see that ths functon s well-defned, note that V (G ) has no ncomng edges (see Fgure 2), so (,j) E(G f(, j) = ) j [λ v +1] j = a. Ths map s a projecton; t s affne and preserves nteger ponts. It s not hard

6 6 KAROLA MÉSZÁROS, CONNOR SIMPSON, AND ZOE WELLNER to see that ϕ s a bjecton between F G (a) and a λ. Furthermore, the second clam of Lemma 3.2 s mmedate. Lemmas 3.1 and 3.2 mply that we can consder any one of F G(λ,n) (a) wth n l(λ) + λ 1 as the lmtng polytope F(λ,a) lm. Indeed, when n λ 1 + l(λ), t s guaranteed that the Young dagram of λ wll ft n the top rght quadrant of an n n matrx. Fgure 2 llustrates the effects of ths Fgure 2. The Young dagram of λ = (3, 2, 1) n both 6 6 and 7 7 matrces, and the correspondng graphs G(λ, 6) and G(λ, 7), wth edges to the snk dotted. Observe that ncreasng n by 1 adds a sngle new vertex wth a sngle outgong edge to the snk. Ths underles the fact that F G(λ,6) (a) and F G(λ,7) (a) are ntegrally equvalent and have the same volume. It also justfes our use of a as the netflow vector for both F G(λ,6) and F G(λ,7) : only the frst l(λ) entres of the netflow vector matter. The decomposton of the lmtng polytope nto smplces also gves us a neat formula for ts volume. Theorem 3.3. Let λ be a partton, n λ 1 + l(λ), and a (Z >0 ) n a vector of postve ntegers. The lmtng polytope of F (λ,a) s ntegrally equvalent to a product of scaled smplces a 1 λ1 a l(λ) λl(λ). Consequently, t has normalze volume (3.1) vol F lm (λ,a) = [l(λ)] λ! [l(λ)] Proof. It s mmedate from Lemmas 3.1 and 3.2 that F(λ,a) lm s ntegrally equvalent to [l(λ)] a λ. The unnormalzed volume of a λ s aλ λ!, so the unnormalzed volume of F lm (λ,a) s l(λ) a λ λ!. a λ λ! To normalze ths volume, we dvde t by the

7 FLOW POLYTOPES OF PARTITIONS 7 volume of the standard smplex of dmenson dm F lm λ, so our endng expresson s (λ,a). F lm (λ,a) has dmenson vol F lm (λ,a) = l(λ) λ! l(λ) a λ λ! where vol F(λ,a) lm lm denotes the normalzed volume of F(λ,a). We note that we can relax the requrement a (Z >0 ) n to a (Z 0 ) n and obtan smlar results. Indeed, both Lemmas 3.1 and 3.2 and ther proofs hold verbatm (f a = 0 then F G (a) s a pont). Thus, the analogues of Lemmas 3.1 and 3.2 yeld a volume formula for any F(λ,a) lm, a (Z 0) n. For smplcty, we wll work wth a (Z >0 ) n throughout the paper Constant Term Identtes. Usng the volume formula gven n Theorem 3.3, we can derve a constant term dentty. Let λ be a partton, n an nteger such that n λ for all [l(λ)], and a Z n >0. For convenence, let L = [l(λ)] λ and let G be the restrcton of G(λ, n) to the vertex set [n]. Further, let λ = (λ 1,..., λ l(λ), 0,..., 0, 0) Z n. Theorem 3.4. CT xn... CT x1 (a 1 x 1 + +a n x n ) L [l(λ)] n+1 λ j n Proof. By Equaton (2.5), the volume of F(λ,a) lm s equal to vol F(λ,a) lm = ( ) L 1,..., n L l()=n j [n] (x x j ) 1 ] = a j j K G ( λ). [l(λ)] Now, let G be G wth all ts edges reversed and observe that K G ( λ) = K G ( λ ). Thus, the above s equal to = ( ) L 1,..., n L l()=n = ( ) L 1,..., n L l()=n ( = CT xn... CT x1 L l()=n j [n] j [n] a j j a j j L K G ( λ ) 1,..., n [x λ ] ) (,j) E(G) j [n] a j j (1 x j x 1 x λ ) 1 (,j) E(G) λ! (1 x j x 1 ) 1. [l(λ)] a λ λ!

8 8 KAROLA MÉSZÁROS, CONNOR SIMPSON, AND ZOE WELLNER Snce the th vertex of G has λ edges out of t, (,j) E(G) (1 x jx 1 ) 1 = x λ (,j) E(G) (x x j ) 1. It follows that the above s equal to ( ) L = CT xn... CT x1 a j j x (x x j ) 1 1,..., n L l()=n j [n] = CT xn... CT x1 (a 1 x a n x n ) L (,j) E(G) (,j) E(G) (x x j ) 1 where the latter equalty follows by the Multnomal Theorem. The product n the last expresson can be rewrtten as [l(λ)] n+1 λ (x j n x j ) 1. Fnally, substtutng n the formula for vol F(λ,a) lm gven n Theorem 3.3 yelds the result. 4. The face structure of polytopes n F (λ,a) In Theorem 3.3, we showed that for all λ and a Z n >0, F(λ,a) lm s ntegrally equvalent to a product of smplces, mplyng that ts combnatoral type s that of a product of smplces. In ths secton, we show that each element of the famly F(λ,a) lm lm has the same combnatoral type as F(λ,a) A quck revew of results relatng subgraphs and the face lattce. Before proceedng, we wll revew some facts relatng the face lattce of a flow polytope to subgraphs of the graph from whch t arrses. Let G be a graph and a a netflow vector. We call a subgraph H of G a-regular (or just regular when the netflow n queston s clear) f there s an a-flow f on G such that f s zero on all edges of G that are not n H. We say that a s n generc poston wth respect to G f there s no a-flow f such that f s the unque flow on two dstnct subtrees of G. The followng two results are mpled by [3, Lemma 2.1 & Theorem 2.2] for the faces of F G(λ,n) (a). Lemma 4.1. The vertces of F G(λ,n) (a) are the flows on the regular subtrees of G (λ,n). Theorem 4.2. If a s n generc poston, then the regular subtrees of G (λ,n) are n bjecton wth the vertces of F G(λ,n) (a) and the faces of F G(λ,n) (a) are n bjecton wth the regular subgraphs of G (λ,n) Characterzaton of regular subtrees. Let λ = (λ 1,..., λ l ) be a partton, n an nteger such that n λ for all [l(λ)], G = G(λ, n), and let a Z n >0. In ths secton, we characterze whch subtrees of G are a-regular for a Z n >0. Lemma 4.3. Let H be a subgraph of G bult by pckng one outgong edge from each vertex < n + 1. Then H s an a-regular spannng tree of G for a Z n >0. Proof. Frst we show that H s acyclc and connected. If there were a cycle C H, then there would have to be two outgong edges from ts mnmal vertex. Thus, H s acyclc. To see that H s connected, we note that our graph has n edges, n + 1 vertces and t has no cycles. To see that H s regular, construct an a-flow on t as follows. Let e v E(H) be the unque edge out of v n H. Let f(e 1 ) = a 1. Assume f(e 1 ),..., f(e ) have been assgned for some 1. Then we let f(e +1 ) = a +1 f + 1 has no ncomng

9 FLOW POLYTOPES OF PARTITIONS 9 edges n H, and f(e +1 ) = a +1 + (v,+1) E(H) f(e v) for vertces wth ncomng edges. Lemma 4.4. Every spannng subtree T of G that admts an a-flow for a Z n >0 has a sngle edge out of each of ts vertces v < n + 1. Proof. Suppose that a spannng subtree T of G has at least two outgong edges from a vertex v. Snce T has n edges and n + 1 vertces, t follows then that T has two vertces wth no outgong edges. In partcular, there s a v < n + 1 wth only ncomng edges. Snce a v > 0, such a tree cannot admt an a-flow. Thus each spannng subtree T of G that admts an a-flow has at most one edge out of each of ts vertces v < n + 1. Snce we need n edges, t has exactly one edge out of each of ts vertces v < n + 1. Theorem 4.5. The a-regular subtrees for a Z n >0 of G(λ, n) are precsely those that have exactly one edge out of every vertex. The polytope F G(λ,n) (a) has l(λ) =1 (λ + 1) vertces, ndependent of n, correspondng to the a-flows on the aforementoned subtrees. Proof. The frst statement follows mmedately from Lemmas 4.3 and 4.4. We can count such trees by notng that each vertex has 1 + λ edges out of t (and 1 edge f > l), so there are l =1 (λ + 1) ways to choose such a tree. Now, note that f T s a spannng tree of G that admts a flow f T, then f T must be nonzero on every edge of T because a s non-snk entres are all postve. Thus, a s n generc poston and by Lemma 4.2 there s a bjecton between subtrees of G that admt regular flows and vertces of F G (a) The face lattce. We are ready to show that every polytope n F (λ,a) has a face lattce somorphc to that of F lm (λ,a). Lemma 4.6. Let λ be a partton, n an nteger such that n λ for all [l(λ)], and a Z n >0. The regular subgraphs of G(λ, n) are precsely those that have at least one edge out of every non-snk vertex. Furthermore, for H and K regular subgraphs of G(λ, n), F K (a) F H (a) f and only f K s a subgraph of H. Thus, the face lattce of F G(λ,n) (a) s somorphc to the poset of regular subgraphs of G(λ, n). Proof. The entres of a are all postve, so every vertex of a regular subgraph must have at least one outgong edge. Conversely, any subgraph that has at least one edge out of every non-snk vertex contans a regular subtree by Theorem 4.5 and s therefore regular. For the second statement, the f mplcaton s clear. For the only f, observe that f e s an edge n K that s not n H, then there s a regular subtree T contaned n K such that e E(T ). Snce a s n generc poston, the unque flow f on T s nonzero on e and s therefore not n F H (a), so F K (a) F H (a). The last statement then follows by Lemma 4.2 snce a s n generc poston. Theorem 4.7. Let λ = (λ 1,..., λ k ) be a partton, n an nteger such that n λ for all [l(λ)], and a Z n >0 a netflow vector. The face posets of the polytopes belongng to F (λ,a) are somorphc to the face poset of the Cartesan product of smplces λ1 λ2 λk. In partcular, the h-polynomal of the polytopes belongng to F (λ,a) s

10 10 KAROLA MÉSZÁROS, CONNOR SIMPSON, AND ZOE WELLNER k =1 λ =0 h x = k [λ ] x = =1 k λ 1 Proof. Let λ = (λ 1 + 1, λ 2 + 1,..., λ k + 1) and let Y be the Young dagram of λ. Let C be the poset of subsets C of the boxes of Y such that C contans at least one box from every row of Y, ordered by ncluson. For any nteger n such that n λ for all [l(λ)], defne the bjecton ϕ n : Y E(G(λ, n)) by ϕ n (, j) = (, n + 2 j). Now, let R(G(λ, n)) be the set of regular subgraphs of G and defne φ n : C R(G(λ, n)) by C =1 j=0 x j. ( [n + 1], ϕ n (C) {(, n + 1) : n + 1 > > l(λ)} Every subgraph n the mage of φ n has an edge out of every vertex besdes n + 1, so by Lemma 4.6, the mage of φ n les n R(G(λ, n)) as clamed. In fact, usng the surjectvty of ϕ n, the mage of φ n s all of R(G(λ, n)). Fnally, njectvty of φ n follows from the fact that ϕ n s njectve, and t s clear that t preserves ncluson. Therefore, φ n s an order preservng bjecton between C and R(G(λ, n)). Applyng the second statement of Lemma 4.6, we have that for any nteger n such that n λ for all [l(λ)], the face lattce of F G(λ,n) (a) s somorphc to C. In partcular, the face lattce of F(λ,a) lm = F G(λ,λ 1+l(λ))(a) s somorphc to C. The former s somorphc to the face lattce of λ1 λ2 λk by Theorem 3.3. It follows that every element of F (λ,a) has a face lattce somorphc to ths one. Furthermore, F (λ,a) s of dmenson k =1 λ and ts h-polynomal s gven by the products of the h-polynomals of the smplces: h(f (λ,a), x) = k =1 h( λ, x) = k =1 [λ ] x, as desred. Remark 4.8. Snce F Kn+1 (1) s an element of F ((n 1,n 2,...,1),1), Theorem 1.2 s a specal case of Theorem 4.7. References [1] W. Baldon and M. Vergne. Kostant parttons functons and flow polytopes. Transform. Groups, 13(3-4): , [2] C.S. Chan, D.P. Robbns, and D.S. Yuen. On the volume of a certan polytope. Experment. Math., 9(1):91 99, [3] L. Hlle. Quvers, cones and polytopes. Lnear Algebra Appl., (365): , 8 [4] K. Mészáros and A. H. Morales. Flow polytopes of sgned graphs and the Kostant partton functon. Int. Math. Res. Notces, (3): , , 2 [5] K. Mészáros, A.H. Morales, and B. Rhoades. The polytope of Tesler matrces. Selecta Mathematca, to appear, [6] R. P. Stanley. Enumeratve combnatorcs. Vol. 2, volume 62 of Cambrdge Studes n Advanced Mathematcs. Cambrdge Unversty Press, Cambrdge, Wth a foreword by Gan-Carlo Rota and appendx 1 by Sergey Fomn. 1 Department of Mathematcs, Cornell Unversty, Ithaca NY E-mal address: karola@math.cornell.edu Department of Mathematcs, Cornell Unversty, Ithaca NY E-mal address: cgs93@cornell.edu Department of Mathematcs, Cornell Unversty, Ithaca NY E-mal address: zaw5@cornell.edu ).

Math Homotopy Theory Additional notes

Math Homotopy Theory Additional notes Math 527 - Homotopy Theory Addtonal notes Martn Frankland February 4, 2013 The category Top s not Cartesan closed. problem. In these notes, we explan how to remedy that 1 Compactly generated spaces Ths